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Chapter 1
Introduction
1.1 Extreme Events
This book is about how best to construct investment portfolios if a priori it is reasonable to assume that markets might exhibit fat-tailed behaviour. It is designed to appeal to a wide variety of practitioners, students, researchers and general readers who may be interested in extreme events or portfolio construction theory either in isolation or in combination. It achieves this aim by
(a) Exploring extreme events, why they might arise in a financial context and how we might best analyse them.
(b) Separately exploring all the main topics in portfolio construction theory applicable even in the absence of fat tails. A special case of any more general approach capable of effectively handling extreme events is the situation where the extent of fat-tailed behaviour is too small to be discernible.
(c) Blending points (a) and (b) together to identify portfolio construction methodologies better able to cater for possible fat-tailed behaviour in the underlying assets or liabilities.
Given its wide intended audience, the book covers these topics both from a more qualitative perspective (particularly in the earlier and later chapters) and from a more quantitative (i.e., mathematical) perspective (particularly in the middle chapters). Where possible, this book has been segmented so that valuable insights can be gained without necessarily having to read the whole text. Conversely, in the author’s opinion, valuable insights arise throughout the book, including the parts that are more mathematical in nature. More general readers are therefore encouraged not to skip over these parts completely, although they do not need to worry too much about following all the details.
By fat-tailed behaviour we mean that the distribution of future returns is expected to involve more extreme events than might be expected to occur were returns to follow the (multivariate) (log-) Normal distributions often assumed to apply to markets in basic portfolio construction texts.1
Most practitioners believe that most markets are ‘fat-tailed’ given this terminology. There is a wide body of empirical academic literature that supports this stance, based on analysis of past market behaviour. There is also a growing body of academic theory, including some involving behavioural finance, explaining why fat-tailed behaviour seems so widespread. So, we might also characterise this book as exploring how best to construct investment portfolios in the real world.
Of course, practitioners and academics alike are not themselves immune from behavioural biases. It is one thing to agree to pay lip service to the notion that market behaviour can be fat-tailed, but quite another to take this into account in how portfolios are actually constructed. Following the dot.com boom and bust in the late 1990s and early 2000s, markets settled into a period of unusually low volatility. Strategies that benefited from stable economic conditions, e.g., ones that followed so-called ‘carry’ trades or strategies relying on continuing ready access to liquidity, proved successful, for a time. The 2007–09 credit crisis, however, painfully reminded the complacent that markets and economies more generally can and do experience extreme events.
1.2 The Portfolio Construction Problem
We do not live in a world in which we have perfect foresight. Instead, portfolio construction always involves striking a balance between risk and reward, i.e., the risk that the views implicit in our portfolio construction will prove erroneous versus the rewards that will accrue if our views prove correct. Everyone involved in the management of portfolios, whether of assets or of liabilities, faces a portfolio construction problem. How do we best balance risk and return? Indeed, what do we mean by ‘best’?
Given the lack of perfect foresight that all mortals face, it is not reasonable to expect a book like this to set out guaranteed ways of profiting from investment conditions come what may. Instead, it seeks to achieve the lesser but more realistic goal of exploring the following:
(a) core elements of portfolio construction;
(b) mathematical tools that can be used to assist with the portfolio construction problem, and their strengths and weaknesses;
(c) ways of refining these tools to cater better for fat-tailed market behaviour;
(d) mindsets best able to cope well with extreme events, and the pitfalls that can occur if we do not adopt these mindsets.
1.3 Coping with Really Extreme Events
Lack of perfect foresight is not just limited to a lack of knowledge about exactly what the future holds. Typically in an investment context we also do not know how uncertain the future will be. Using statistical terminology, we do not even know the precise form of the probability distribution characterising how the future might evolve.
The differentiation between ‘risk’ and ‘uncertainty’ is a topic that several popular writers have sought to explore in recent times, e.g., Taleb (2004, 2007). In this context ‘risk’ is usually taken to mean some measurable assessment of the spread of possible future outcomes, with ‘uncertainty’ then taken to mean lack of knowledge, even (or particularly) concerning the size of this spread.
In this book, we take up this baton particularly in Chapters 8 and 9. Holding such an insight in mind is, I think, an important contributor to successful portfolio construction. In particular, it reminds us that really extreme events seem to have a nasty habit of occurring more often than we might like. Put statistically, if there is a 1 in 1010 (1 in 10 billion) chance of an event occurring given some model we have adopted, and there is a 1 in 106 (1 in a million) chance that our model is fundamentally wrong, then any really extreme events are far more likely to be due to our model being wrong than representing random (if unlikely) draws from our original model.2
Yet such insights can also be overplayed. The portfolio construction problem does not go away merely because the future is uncertain. Given a portfolio of assets, someone, ultimately, needs to choose how to invest these assets. Although it is arguably very sensible for them to bear in mind intrinsic limitations on what might be knowable about the future, they also need some framework for choosing between different ways of structuring the portfolio.
This framework might be qualitatively formulated, perhaps as someone’s ‘gut feel’. Alternatively, it might be quantitatively structured, based on a more mathematical analysis of the problem at hand. It is not really the purpose of this book to argue between these two approaches. Indeed, we shall see later that the outcome of essentially any qualitative judgemental process can be reformulated as if it were coming from a mathematical model (and arguably vice versa).
Perhaps the answer is to hold onto wealth lightly. All of us are mortal. The more religious among us, myself included, might warm to this philosophy. But again, such an answer primarily characterises a mindset to adopt, rather than providing specific analytical tools that we can apply to the problem at hand.
1.4 Risk Budgeting
Some practitioners point to the merits of risk budgeting. This involves identifying the total risk that we are prepared to run, identifying its decomposition between different parts of the investment process and altering this decomposition to maximise expected value-added for a given level of risk. It is a concept that has wide applicability and is difficult to fault. What business does not plan its evolution via forecasts, budgets and the like? Indeed, put like this risk budgeting can be seen to be common sense.
Again, though, we have here principally a language that we can use to describe how to apply investment principles. Risk budgeting principally inhabits the ‘mindset’ sphere rather than constituting an explicit practical toolset directly applicable to the problem at hand. This should not surprise us. Although sensible businesses clearly do use budgeting techniques to good effect, budgeting per se does not guarantee success. So it is with risk budgeting.3
However, language is the medium through which we exchange ideas and so cannot be ignored. Throughout this book, we aim to explain emerging ideas using terms that can be traced back to risk budgeting concepts. This helps clarify the main aspects of the methodology under discussion. It also helps us understand what assumptions need to be made for the relevant methodology to be valid.
1.5 Elements Designed to Maximise Benefit to Readers
As explained in Section 1.1, this book aims to appeal to a wide variety of audiences. To do this, I have, as with my earlier book on Market Consistency, sought a suitable balance between mathematical depth and readability, to avoid some readers being overly daunted by unduly complicated mathematics. The book focuses on core principles and on illuminating them where appropriate with suitably pitched mathematics. Readers wanting a more detailed articulation of the underlying mathematics are directed towards the portfolio construction pages of the www.nematrian.com website, referred to throughout this book as Kemp (2010).
To maximise the benefit that both practitioners and students can gain from this book, I include two sections at the end of each main chapter that provide:
(a) Comments specifically focusing on the practitioner perspective. To navigate successfully around markets typically requires an enquiring yet somewhat sceptical mindset, questioning whether the perceived benefits put forward for some particular technique really are as strong as some might argue. So, these sections either focus on the ways that practitioners might be able to apply insights set out earlier in the relevant chapter in their day-to-day work, or highlight some of the practical strengths and weaknesses of techniques that might be missed in a purely theoretical discussion of their attributes.
(b) A discussion of some of the more important implementation challenges that practitioners may face when trying to apply the techniques introduced in that chapter. Where the same challenge arises more than once, I generally discuss the topic at the first available opportunity, unless consideration of the challenge naturally fits better in a later chapter.
The book also includes an Appendix containing some exercises for use by students and lecturers. Each main chapter of the book has associated exercises that further illustrate the topics discussed in that chapter. The exercises are reproduced with kind permission from Nematrian Limited. Hints and model solutions are available on the www.nematrian.com website, as are any analytical tools needed to solve the exercises.
Throughout the book, I draw out principles (i.e., guidance, mainly for practitioners) that have relatively universal application. Within the text these principles are indented and shown in bold, and are referenced by P1, P2, etc.
1.6 Book Structure
The main title of this book is Extreme Events. It therefore seems appropriate to focus first, in Chapters 2 and 3, on fat tails and extreme events. We explore some of the ways in which fat-tailed behaviour can be analysed and the existence or otherwise of extreme events confirmed or rejected. We differentiate between analysis of fat tails in single return series in Chapter 2 and analysis of fat tails in joint (i.e., multiple) return series in Chapter 3. The shift from ‘one’ to ‘more than one’ significantly extends the nature of the problem.
Before moving on to portfolio construction per se, we consider in Chapter 4 some ways in which we can identify what seems to be driving market behaviour. Without some underlying model of market behaviour, it is essentially impossible to assess the merits of different possible approaches to portfolio construction (or risk modelling). We consider tools such as principal components analysis and independent components analysis, and we highlight their links with other statistical and econometric tools such as multivariate regression.
In Chapters 5--7 we turn our attention to the portfolio construction problem.
Chapter 5 summarises the basic elements of portfolio construction, both from a quantitative and from a qualitative (i.e., ‘fundamental’) perspective, if fat tails are not present. At a suitably high level, both perspectives can be viewed as equivalent, apart perhaps from the mindset involved. In Chapter 5 we also explore some of the basic mathematical tools that commentators have developed to analyse the portfolio construction problem from a quantitative perspective. The focus here (and in Chapter 6) is on mean-variance portfolio optimisation (more specifically, mean-variance optimisation assuming time stationarity). We consider its application both in a single-period and in a multi-period world.
In Chapter 6 we highlight the sensitivity of the results of portfolio construction analyses to the input assumptions, and the tendency of portfolio optimisers to maximise ‘model error’ rather than ‘risk-return trade-off’. We explore ways of making the results more robust to errors affecting these input assumptions. The academic literature typically assumes that input assumptions are estimated in ...
