June 2014


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imageTakeaway :

Most of the phenomena in our world are periodic in nature. Yet, econometric courses at undergraduate / graduate level inevitably start  from the time domain instead of frequency domain. This book is a great introduction for some one looking to get an overview of frequency domain analysis. All the principles are explained from a regression standpoint  in the initial chapters. Discrete Fourier Transforms are gradually introduced to connect various ideas. Both univariate and multivariate time series are dealt at length with just enough mathematical rigor.

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In today’s world, parents are extremely observant about how their children are learning. Be it academics or music or sport any other field that the child has developed a semblance of liking, the parent gives and seeks all the guidance available to make his/her kid’s learning process effective. Given the hyperconnected instant gratification world that we are all living it, Kids left to their own devices, become just that, in the literal sense. Their lives are surrounded by world of devices (cell phones, gaming consoles, ipod, ipad, etc.) and naturally they develop an affinity towards them. One doesn’t need some academic research to infer that attention spans are going down across all age groups, more so in children. In such an environment, can parents or teachers be confident that the children develops thinking and meta-thinking(thinking about how they are thinking )skills to become effective learners ?.

There is a mad rush towards alternative education schools everywhere. Parents are under the notion that schools that focus on standardized testing and standardized learning might not be effective for their kid whom they think is somehow “special” from everyone. In what sense they are “special”, only future would tell, but that doesn’t stop them from thinking that education must be somehow customized to suit their kid’s learning style.

In my own family, I have seen my cousin’s kids being put through a school where there are no tests at all until 8th or 9th grade. The school advertises to the general public saying that their USP is small class room sizes and NO TESTS. The admission process in the school creates a massive frenzy amongst everyone and even gets cited in the local newspapers. Parents feel that this “NO TEST” environment will unleash creativity amongst their kids and turn their little ones in to a creative genius. Is it really true that an environment without tests fosters good learning? Why is there a universal backlash against “tests”? Why is everyone fixated on “learning styles”? What is wrong with the current educational system? How does one become an effective learner? These and many more questions are answered in this book. Here is an attempt to summarize the book.

Learning Is Misunderstood

This chapter is a prelude to the book and lists down the claims that the authors verify via field research in various chapters of the book. What are the claims made at the outset?

  • Learning is deeper and more durable when it’s effortful. Learning that’s easy is like writing in sand, here today and gone tomorrow.
  • We are poor judges of when we are learning well and when we’re not. When the going is harder and slower and it doesn’t feel productive, we are drawn to strategies that feel more fruitful, unaware that the gains from these strategies are often temporary.
  • Rereading text and massed practice of a skill or new knowledge are by far the preferred study strategies of learners of all stripes, but they’re also among the least productive. By massed practice we mean the single-minded, rapid-fire repetition of something you’re trying to burn into memory, the “practice-practice-practice” of conventional wisdom. Cramming for exams is an example . Rereading and massed practice give rise to feelings of fluency that are taken to be signs of mastery, but for true mastery or durability these strategies are largely a waste of time.
  • Retrieval practice—recalling facts or concepts or events from memory— is a more effective learning strategy than review by rereading. Periodic practice arrests forgetting, strengthens retrieval routes, and is essential for hanging onto the knowledge you want to gain.
  • When you space out practice at a task and get a little rusty between sessions, or you interleave the practice of two or more subjects, retrieval is harder and feels less productive, but the effort produces longer lasting learning and enables more versatile application of it in later settings.
  • Trying to solve a problem before being taught the solution leads to better learning, even when errors are made in the attempt.
  • People do have multiple forms of intelligence to bring to bear on learning, and you learn better when you “go wide,” drawing on all of your aptitudes and resourcefulness, than when you limit instruction or experience to the style you find most amenable.
  • When you’re adept at extracting the underlying principles or “rules” that differentiate types of problems, you’re more successful at picking the right solutions in unfamiliar situations. This skill is better acquired through interleaved and varied practice than massed practice.
  • In virtually all areas of learning, you build better mastery when you use testing as a tool to identify and bring up your areas of weakness.
  • Elaboration is the process of giving new material meaning by expressing it in your own words and connecting it with what you already know. The more you can explain about the way your new learning relates to your prior knowledge, the stronger your grasp of the new learning will be, and the more connections you create that will help you remember it later.
  • Rereading has three strikes against it. It is time consuming. It doesn’t result in durable memory. And it often involves a kind of unwitting self-deception, as growing familiarity with the text comes to feel like mastery of the content.
  • It makes sense to reread a text once if there’s been a meaningful lapse of time since the first reading, but doing multiple readings in close succession is a time-consuming study strategy that yields negligible benefits at the expense of much more effective strategies that take less time. Yet surveys of college students confirm what professors have long known: highlighting, underlining, and sustained poring over notes and texts are the most-used study strategies, by far.
  • Rising familiarity with a text and fluency in reading it can create an illusion of mastery. As any professor will attest, students work hard to capture the precise wording of phrases they hear in class lectures, laboring under the misapprehension that the essence of the subject lies in the syntax in which it’s described. Mastering the lecture or the text is not the same as mastering the ideas behind them . However, repeated reading provides the illusion of mastery of the underlying ideas. Don’t let yourself be fooled. The fact that you can repeat the phrases in a text or your lecture notes is no indication that you understand the significance of the precepts they describe, their application, or how they relate to what you already know about the subject.

All the above claims are verified by experiments carried out in various school settings and other unconventional places. The authors at the very beginning make it clear that the learning theories that have been handed down to us have been a result of theory, lore and intuition. But over the last forty years and more, cognitive psychologists have been working to build a body of evidence to clarify what works and to discover the strategies that get results.

To Learn Retrieve

We all forget things. If they are trivial stuff, they really don’t matter. But if they are key principles, concepts, then our learning will be stunted and it becomes painfully obvious that we need to re-read the forgotten stuff. To give a specific example, let’s say I am learning about Jump modeling and there is an introductory section on Poisson processes. In the past I would have spent some time going over Poisson processes and understanding the math behind it. The key theorems are somewhere in my memory. Not all are at my beck and call. So, whenever I come across a concept that I have tough time recalling, my usual strategy is to re-read the old section. Not an ideal strategy, says this chapter. I think most of us follow the above strategy where re-reading is the goto choice to make things fresh. The chapter focuses on one key point – retrieval practice. This is a kind of practice where you make an effort to recall those concepts from your memory, reflect on those concepts from time to time. This is not the same as rereading the text.

The authors make a strong case for “testing” as a means of retrieval practice. The retrieval effect in the cognitive psychology field is known as “testing effect”. Through the results of various experiments conducted, the authors suggest that testing immediately after a lecture, or testing yourself at spaced intervals is far better than rereading at spaced intervals. Repeated retrieval ties the knot of memory. Retrieval must be spaced out rather than becoming a mindless repetition. It should require cognitive effort. The authors back up these suggestions with field experiments that show that frequent testing of students with delayed feedback gave better performance than merely rereading or revisiting the material before midterm and end term.

For an adult learner, how does this apply? I guess one must self test, even though it is painful. It is actually better if it is painful as it leads to greater effort at retrieval and hence better learning. How should the tests be designed? For programming there are many suggestions out there. For something like math, I think the best way is to read a theorem and try to give a proof in your own words trying to recall whatever you have learnt the previous time around. Merely reading through the proofs or concepts will not make the learning stick. This is called the “generation effect “. You generate the proof from some clues. In the case of frameworks or set of ideas, you can probably write an essay recalling all the aspects of the theory without rereading. It is easy to fall in to the trap of ,” Ok I have forgotten, let me reread the material”. Instead this chapter says that one must pause, take a self test, then quiz yourself as you go over the material again, and then reflect on what you have relearnt. This has a term called “elaboration” in the literature. This means you elaborate the learning or practice session so that memory paths are strengthened. The other thing I have started following recently is to take a small 60 page booklet and keep noting down whatever you find interesting for the day in the form of statements, visuals or just about anything that captures the learning. Obviously as you go along these 60 page diaries accumulate. Once in a while you can pick a booklet and read the statements that you found interesting a month ago, 6 months ago, a year ago. This is a kind of retrieval practice where you are trying to learn better by testing yourself.

 

Mix up practice

The authors introduce a term called “mass practice” where you keep practicing one aspect of skill development until you are good at it and then move on. This is clearly seen in textbooks where each chapter is followed by a set of problems that are only relevant to that chapter and the reader is asked to practice the exercises and then move on to the next chapter. This is the usual advice passed on to us from many people. Practice, practice until the skill is burned in to the memory. Faith in focused repetitive practice is everywhere and this is what the authors mean by “mass practice” Practice that is spaced out, interleaved with other learning, and varied produces better mastery, longer retention and more versatility. There is one price to pay though. From the part of the learner, it requires more effort. There are no quick positive affirmations that come with mass practice. Let’s say you have been immersed in doing Bayesian analysis for a few months, then you take a break and get back to it, there will be an inherent slowness at which you can digest things as those concepts are lying somewhere in your long term memory and invoking them takes effort. But the authors say that this is a good thing. Even though learning feels slower, this is the way to go.

This phenomenon of “mass practice” is everywhere. Summer camps, focused workshops, training seminars. Spacing out your practice feels less productive for the very reason that some forgetting has set in and you’ve got to work harder to recall concepts. It doesn’t feel like you’re on top of it. What you don’t sense in the moment is that the added effort is making the learning stronger.

Why does spaced practice work? Massed practice is good for short term memory. But the sad part is that such a practice does not lead to durable learning. For something to get in to long term memory, there should be consolidation in which memory traces are strengthened. If you do not activate these memory traces, then the paths will be lost. It is like laying a new road and using it for a week or so, and then moving on. Unless you use the road often, the material never gets a chance to become strong and in the process loses vitality.

Practice also has to interleaved. Interleaving is practicing two or more subjects or two different aspects of the same subject . You cannot study one aspect of subject completely and move on to another subject and so on. Linearity isn’t good. Let’s say you are learning some technique, for example EM algorithm. If you stick to data mining field, you will see its application in let’s say mixture estimation. However by interleaving your practice with let’s say state space models, you see that EM algorithm being used to estimate hyperparameters of a model. This interleaving of various topics gives a richer understanding. Obviously there is a price to pay. The learner is just about learning to understand something, when he is asked to move to another topic. So, that sense of feeling that he hasn’t got a full grasp on the topic remains. It is a good thing to have but an unpleasant situation that a learner must handle.

Varied practice – Let’s say you are a quant and trying to build financial models. Varied practice in your case would be to build a classification model, a Bayesian inference model, a Brownian motion based model, a more generic Levy process based model, a graph based model etc. The point is that you develop a broader understanding of relationships between various aspects of model building. If you stick to let’s say financial time series for an year, then move on to machine learning for another year, it is likely that you are going to miss connections between econometric models and machine learning models. Having said that, it is not a pleasant feeling to incorporate varied practice in one’s schedule. Imagine you are just about understand the way to build a particle filter, a technique to do online estimation of state vector and you have already spent quite an amount of time on that subject. It is time to move to another area, let’s say building a levy process based model. As soon as you start doing something on Levy process, you sense that your knowledge Poisson + Renewal processes is very rusty and the learning is extremely slow. This is the unpleasant part. But the authors have a reassuring message. When the learning appears slow and effortful, that is where real learning is taking place.

Compared to mass practice, a significant advantage of interleaving practice is that they help us learn to assess context and discriminate between problems, selecting and applying problems from a range of possibilities. The authors give an example of “learning painting styles” to drive home the point of interleaved and varied practice.

Practice like you play and you will play like you practice. The authors stress the importance of simulations for better practice. If you are in to trading strategy development, simulating time series and testing the strategy out of sample is fundamental for a better understanding of the strategy. In fact, with the rise of MCMC, the very process of estimation and model selection is done via simulation. The authors also bring out an example where daily reflection can be done as a form of retrieval practice.

I liked the last section of this chapter where the authors share the story of Georgia university football coach who is following the principles of retrieval, spacing, interleaving, variation, reflection and elaboration, in making his college team a better playing team.

 

Embrace Difficulties

Short term impediments that make for stronger learning have come to be called desirable difficulties, a term coined by Elizabeth and Robert Bjork. The chapter starts with an example of military school where the trainees are not allowed to carry note books or write stuff. They have to listen, watch, rehearse and execute. Testing is a potent reality check on the accuracy of your own judgment of what you know how to do. The process of strengthening the long term memory is called consolidation. Consolidation and transition of learning to long-term storage occurs over a period of time. An apt analogy for how the brain consolidates new learning is the experience of composing an essay. Let’s say you are studying point processes, a class of stochastic processes. First time around you might not be able to appreciate all the salient points of the text. You start out feeling disorganized and the most important aspects are not salient. Consolidation and retrieval helps solidify these learning’s. If you are practicing over and over again in some rapid-fire fashion, you are leaning on short term memory and very little mental effort is needed. There is an instant improvement, but the improvement is not robust enough to sustain. But if you practice by spacing and interleaving, the learning is much deeper and you will retrieve far easily in the future.

Durable robust learning means we do two things – First, as we recode and consolidate new material from short term memory into long term memory, we must anchor there securely. Second, we must associate it with a diverse set of cues that will make us adept at recalling the material later. Having effective retrieval clues is essential to learning and that is where tools like mindmaps help a lot. The reason we don’t remember stuff is that we don’t practice and apply it. If you are in to building say math/stat models, it is essential to at least simulate some data set, build a toy model so that practice gets some kind of anchorage for retrieval. Without this, any reading of a model will stay in your working memory for some time and then vanish. Knowledge, learning and skills that are vivid, hold significance, and those that are practiced periodically stay with us. Our retrieval capacity is limited and is determined by the context, by recent use, and the number and vividness of the cues that you have linked to the knowledge and can call on to help it bring it forth.

Psychologists have uncovered a curious inverse relationship between ease of retrieval practice and the power of that practice to entrench learning, the easier knowledge or skill is for you to retrieve, the less your retrieval practice will benefit your retention of it.

There is an excellent case study of a baseball team where the team is split in to two and they are given varied practice regimen. First group practices 45 pitches evenly divided in to sets of three where each set has a specific type of pitch thrown. The second group also practices 45 pitches but this time, the pitches were randomly interspersed. After the training, the first group feels good about their practice while the second group feels that they were not developing their skills properly. However when it came to the final performance test, the second group performed far better than the first group. This story illustrates two points – first, our judgments of what learning strategies work best for us are often mistaken, colored by illusions of mastery. Second, some difficulties that require more effort and slow down apparent gains will feel less productive but will more than compensate for that by making the learning stronger, precise and durable. The more you have forgotten a topic, the more effective relearning will be in shaping your permanent knowledge. The authors also make it a point to highlight that if you struggle to solve a problem before being shown how to solve it, the subsequent lesson is better learned and more durably remembered.

This chapter and this book is an amazing fountainhead of ideas that one can use. Not everything is new but the fact that there is an empirical evidence to back it up means that you know that it is not folklore wisdom. One thing I learnt from the book which has reinforced my way of learning is “write to learn”. After reading a book or reading a concept, I try to write it down so that I can relate to things that I have already learnt, relate to aspects of the field that I want to eventually apply etc. This obviously takes up a lot of time, but the learning is far more robust. I think book summaries that I manage to write is one of the best ways to reflect on the main contents of the book. I tend to write a pretty detailed summary of key ideas so that the summary serves as a material for retrieval practice at a later point in time.

The other idea this chapter talks about is about the need to commit errors to solidify learning. Came to know about “Festival of errors” and “Fail conference”. There is also a story about Bonnie, a writer and self-taught ornamental gardener, who follows the philosophy, “ leap before you look because if you look, you probably won’t like what you see”. Her garden writing appears under the name “Blundering Gardener”. Bonnie is a successful writer and her story goes on to show that struggling with a problem makes for stronger learning and how a sustained commitment to advancing in a particular field of endeavor by trial-and-error leads to complex mastery and greater knowledge of interrelationships of things. Bonnie’s story is pretty inspiring for anyone who wishes to tackle a difficult field. By going head long in to the field and learning from the trial and error process, and then writing about the entertaining snafus and unexpected insights, she is doing two things. Firstly, she is retrieving the details and elaborating the details. “Generative learning” means learner is generating the answer than recalling it. Basically it means learn via trial-and-error.

Takeaways

  • Learning is a three step process – Initial encoding, consolidation and retrieval
  • Ability to recall what you already have depends on the repeated use of information and powerful retrieval clues
  • Retrieval practice that’s easy does little to strengthen learning, the more difficult the practice, the greater the benefit
  • Retrieval needs to be spaced. When you recall something from your memory when it has already become rusty, you need more effort and this effortful retrieval strengthens memory and makes learning pliable
  • Practice needs to be interleaved and varied
  • Trying to come up with an answer rather than presenting it to you leads to better learning and retention


Avoid Illusions of knowing

The chapter starts by describing two modes of thinking System 1 and System 2, from Daniel Kahneman’s book and says that we base our actions based on System 1 more often than System 2. Our inclination to finding narratives means that it has a significant say in our memory capabilities. There are lot of illusions and misjudgments that we carry along. One way to escape from them is to replace subjective experience as the basis for decisions with a set of objective gauges outside ourselves, so that our judgments squares with the real world around us. When we have reliable reference points, we can make good decisions about where to focus our efforts, recognize where we’ve lost our bearings, and find out way again. It is important to pay attention to the cues you are using to judge what you have learned. Whether something feels familiar or fluent is not always a reliable indicator of learning. Neither is your level of ease in retrieving a fact or phrase on a quiz shortly after the text. Far better is to create a mental model of the material that integrates various ideas of the text, connects to what you already know, and enables you to draw inferences. How ably you can explain the text is an excellent cue for judging comprehension, because you must recall the salient points, put in your own words and give the logic of how it connects to everything else.

Get beyond your learning styles

It is a common statement that you come across in the media, “every kid is different, the learning style has to be specific and catering to the kid’s learning style”. On the face of it, the statement looks obvious. Empirical evidence however does not support it. The authors give a laundry list of all the learning styles that have been put forth and say that there is absolutely no evidence that catering to individual learning style makes any difference. The simple fact that different theories embrace such wildly discrepant dimensions gives cause for concern about their scientific underpinnings. While it’s true that most all of us have a decided preference for how we like to learn new material, the premise behind learning styles is that we learn better when the mode of presentation matches the particular style in which an individual is best able to learn. That is the critical claim.

The authors say that

Moreover, their review showed that it is more important that the mode of instruction match the nature of the subject being taught: visual instruction for geometry and geography, verbal instruction for poetry, and so on. When instructional style matches the nature of the content, all learners learn better, regardless of their differing preferences for how the material is taught.

So, if the learning styles don’t matter, how should one go about ? The authors mention two aspects here

  1. Structure building: There do appear to be cognitive differences in how we learn, though not the ones recommended by advocates of learning styles. One of these differences is the idea mentioned earlier that psychologists call structure building: the act, as we encounter new material, of extracting the salient ideas and constructing a coherent mental framework out of them. These frameworks are sometimes called mental models or mental maps. High structure- builders learn new material better than low structure-builders.
  2. Successful intelligence: Go wide: don’t roost in a pigeonhole of your preferred learning style but take command of your resources and tap all of your “intelligences” to master the knowledge or skill you want to possess. Describe what you want to know, do, or accomplish. Then list the competencies required, what you need to learn, and where you can find the knowledge or skill. Then go get it. Consider your expertise to be in a state of continuing development, practice dynamic testing as a learning strategy to discover your weaknesses, and focus on improving yourself in those areas. It’s smart to build on your strengths, but you will become ever more competent and versatile if you also use testing and trial and error to continue to improve in the areas where your knowledge or performance are not pulling their weight.

 

Increase your abilities

This chapter starts off by giving some famous examples like the popular Marshmallow study, Memory athletes to drive home the point that brain is every changing. This obviously means that the authors take the side of nurture in the nature vs. nurture debate. The brain is remarkably plastic, to use the term applied in neuroscience, even into old age for most people. The brain is not a muscle, so strengthening one skill does not automatically strengthen others. Learning and memory strategies such as retrieval practice and the building of mental models are effective for enhancing intellectual abilities in the material or skills practiced, but the benefits don’t extend to mastery of other material or skills. Studies of the brains of experts show enhanced myelination of the axons related to the area of expertise but not elsewhere in the brain. Observed myelination changes in piano virtuosos are specific to piano virtuosity. But the ability to make practice a habit is generalizable. To the extent that “brain training” improves one’s efficacy and self-confidence, as the purveyors claim , the benefits are more likely the fruits of better habits, such as learning how to focus attention and persist at practice.

After an elaborate discussion on IQ, the authors suggest three strategies to amp up the performance levels :

  1. Maintaining a growth mindset — Carol Dweck’s work is used as the supporting argument. Dweck came to see that some students aim at performance goals, while others strive toward learning goals. In the first case, you’re working to validate your ability. In the second, you’re working to acquire new knowledge or skills. People with performance goals unconsciously limit their potential. If your focus is on validating or showing off your ability, you pick challenges you are confident you can meet. You want to look smart, so you do the same stunt over and over again. But if your goal is to increase your ability, you pick ever-increasing challenges , and you interpret setbacks as useful information that helps you to sharpen your focus, get more creative, and work harder.
  2. Deliberate Practice – Well, this has become a common term after many authors have written journalistic accounts of Anders Ericsson’s research. In essence it means that expert performance in medicine, science, music, chess, or sports has been shown to be the product not just of innate gifts, as had long been thought, but of skills laid down layer by layer, through thousands of hours of dedicated practice.
  3. Memory cues – Until a learner develops a deep learning of a subject, he/she can resort to mnemonic devices. Conscious mnemonic devices can help to organize and cue the learning for ready retrieval until sustained, deliberate practice and repeated use form the deeper encoding and subconscious mastery that characterizes expert performance.

It comes down to the simple but no less profound truth that effortful learning changes the brain, building new connections and capability. This single fact— that our intellectual abilities are not fixed from birth but are, to a considerable degree, ours to shape— is a resounding answer to the nagging voice that too often asks us “Why bother?” We make the effort because the effort itself extends the boundaries of our abilities. What we do shapes who we become and what we’re capable of doing. The more we do, the more we can do. To embrace this principle and reap its benefits is to be sustained through life by a growth mindset. And it comes down to the simple fact that the path to complex mastery or expert performance does not necessarily start from exceptional genes, but it most certainly entails self-discipline, grit, and persistence ; with these qualities in healthy measure, if you want to become an expert, you probably can. And whatever you are striving to master, whether it’s a poem you wrote for a friend’s birthday, the concept of classical conditioning in psychology, or the second violin part in Hayden’s Fifth Symphony, conscious mnemonic devices can help to organize and cue the learning for ready retrieval until sustained, deliberate practice and repeated use form the deeper encoding and subconscious mastery that characterize expert performance.

 

Make It Stick

The authors implement the lessons from the book within the confines of the book itself. This chapter is like a spaced repetition of all the ideas mentioned in the previous chapters. So, if you don’t bother about the empirical evidence, you can just read this chapter and take them at face value, incorporate them in your schedule and see if they make sense.

imageTakeaway

This is by far best book I have read that talks about “ how to go about learning something? ”. There are gems in this book that any learner can incorporate in one’s schedule and see a drastic change in their learning effectiveness. The book is a goldmine for students, teachers and life-long learners. I wish this book was published when I was a student!.

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Crisis hits financial markets at regular intervals but the market participants keep assuming that they “understand the behavior” of markets and are in “total control” of the situation until the day things crash. There is an army of portfolio managers, equity research analysts, macro analysts, low frequency quants,derivative modeling quants, high frequency quants etc., all trying to understand the markets and trying to make money out of it. Do their gut /intuitive/quant models come close to how the market behaves ?.

I had bought this book in May 2008 assuming that I will read the book on some weekend. Somehow that weekend never came. Years flew by. For some reason I stumbled on to this book again and decided to read through this book. In the last 6 years I have read many books that have highlighted the inadequacy of quant models to capture market place reality. Before going through the book , I had a question – “How is this book different from all those model bashing books?”.

Here is a list of other questions that I had before reading the book :

  • Gaussian models do not predict the kind of randomness that we see in our markets. I had anticipated that the book is going to be Gaussian bashing book. What’s the alternative to Gaussian models?
  • Some practitioners have included jumps in to the usual Brownian motion based models and have tried to explain various market observed phenomena like option skew, volatility jumps, etc. What does the book have to say about Jump modeling? Are Jump processes good at only explaining the past or Can they serve as a tool for better trading and risk management decisions ?
  • Are fractals a way of looking at price processes?  If so, how can one simulate a price process that is based on fractal geometry ?
  • What are the parameters of a fractal model?
  • How do you fit the parameters of a fractal model?
  • Can one put some confidence bounds on the fractal model parameters? If so, how?
  • Why hasn’t the idea taken off in the financial world, if the fractals are ubiquitous in nature?
  • Why are academics reluctant to take a fractal view of the market?
  • Why are quants reluctant to take a fractal view of the market?
  • What is Mandelbrot’s view of financial modeling and risk management? Can risk in the financial markets be managed at all, given the wild randomness that is inherent in the market ?

This book answers many more questions than those that I have listed above. Let me try to summarize briefly each of the 13 chapters of this book:

Risk, Ruin and Reward:
This chapter gives a preview of the basic idea in the book. Fundamental analysis, technical analysis and the more recent random walk analysis all fail to characterize the market swings. However there has been a massive reluctance to move away from such kind of mental models. Partly the system of incentives is to be blamed. The authors say that the goal of the book is to enable readers appreciate a new way of looking at markets, a fractal view of the market. The background to this view is put forth in terms of five rules of market behavior.

  1. Markets are risky
  2. Trouble runs in streaks
  3. Markets have a personality
  4. Markets mislead
  5. Market time is relative

At the outset the authors say that this is not a “how to get rich” kind of book but more of a pop science book.

By the Toss of a Coin or the Flight of the Arrow:

The authors introduce “types” of randomness in this chapter. There are two types of randomness, “mild” and “wild”, the former is akin to tossing a coin and the second is akin to a drunk archer hitting a target. The former is the Gaussian world where things are normal and significant deviations from the mean are almost 0 probability, whereas the latter is the Cauchy distribution world where averages don’t converge and the deviations appear so big that the term “average” doesn’t make sense. Using these two distributions, the authors give a glimpse in to the spectrum of randomness types possible. Financial research, tools and software in today’s world are predominantly built around with the assumption of mild randomness. However the world seems to be characterized by “wild” randomness.

Bachelier and his Legacy:

The authors dig in to the history and recount Bachelier’s life. He was the first person to apply mathematical principles to stock market. He is credited for viewing the price process as an arithmetic Brownian motion. This view was never appreciated by his contemporaries. Almost 50 years later, Bachelier’s paper was picked up by Samuelson, a professor at MIT. Samuelson was a wizard at mathematics and he was on a course to make economics a quantitative discipline. Samuelson and others like Fama( University of Chicago) worked on Bachelier’s view of the market, improved it and applied to a whole host of areas like portfolio management, risk assessment, etc. Castles were being built on sand and these castles were multiplying too in different locations. Almost no one wanted to verify the basic assumption of continuous Brownian motion process.

The House of Modern Finance:

The authors talk about three pillars of modern finance that is taught all over the universities, i.e. CAPM(Capital Asset Pricing Model), MPT(Modern Portfolio Theory),BSM( Black Scholes and Merton framework). The first was developed by Markowitz, the second by Sharpe and the third by Fischer Black,Myron Scholes and Robert Merton. CAPM dealt with dishing out portfolios to investors based on their risk appetite. MPT simplified CAPM and dished out a market portfolio, that later became the bedrock for index funds, and BSM was a framework for valuing options. The crux of each of the pillar goes back to Bachelier’s work. No one was willing to criticize the component of Brownian motion that was used in all these frameworks. There was a rapid adoption of these tools in the industry, almost blind-faith obedience to all these models. Soon however financial crisis brought semblance amongst investors and traders and some of them began to develop strong dose of skepticism over these models. Warren Buffet, once jested that he would create a fund so that academicians could keep teaching these three pillars of modern finance to people so that he would make money by trading against them.

This book was written in 2004 but did it stop people from using mild randomness models for the market ?. No. The ever wide spread usage of Gaussian copula model before the full fledged onset of subprime crisis is a point in case that the castles of air built on mild randomness are still being used. VIX index is a nice example of how people are shying away from using the volatility based on underlying. VIX uses option prices to get an estimate of ATM volatility. All said and done, it does use GBM. So, it is a “FIX” too.

The Case against Modern Theory of Finance:

The authors use this chapter to set the stage for the next part of the book, i.e. way to understand different types of randomness (mild, wild and everything in between). The following are some assumptions that make all the academic theories that have been developed till date, very fragile. They are

  • People are rational and aim only to get rich
  • All investors are alike
  • Price change is practically continuous
  • Price changes follow a Brownian motion

Look at the daily returns of the Nifty total returns index. They are nowhere near close to that generated by a Geometric Brownian motion process that is assumed in finance literature. The second visual has been generated with the assumption of GBM with observed nifty volatility looks too regular as compared to the reality.

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The above picture compares the frequency counts of returns for GBM path and Nifty. One can see that blue bars are all over the place as compared to GBM counts. A few pictures relating to price changes make it abundantly clear that the returns do not follow a GBM for equity, forex, commodity markets. Since various crisis have hit the markets, many researchers have observed CAPM anomalies such as P/E effect, January effect, Market-to-Book effect that do not make sense in a CAPM world. The authors say that the most saddest thing is that when things do not fit the model, the model is tweaked here and there to explain the market behavior. Some of the tweaks that have been created are

  • Instead of one factor, introduce many factors that explain the returns
  • Instead of constant volatility, incorporate a conditional equation for the volatility

The authors vehemently argue that these are all adhoc fixes. All are variations around Gaussian model and hence are fundamentally flawed.

Turbulent Markets: A Preview

The authors make a case for using “turbulence” as a way to study price processes. This chapter gives a few visuals that can be generated via multi fractal geometry.

Studies in Roughness:

The authors use extensive set of images to convey the basic properties of a fractal. A fractal can be identified based on three characteristics

  1. Initiator – Classical geometric object , i.e. a straight line, triangle, solid ball etc.
  2. Generator – A template from which fractal will be made.
  3. Rule of Recursion – This decides the way fractal is repeated

The authors generalize the Euclidean dimension and introduce fractional dimension. This concept is made clear via examples and pictures. The concept of dimension is based on the shrinking ruler length and the increasing measurement length. In most of the phenomena that authors have researched, they have found a surprising convergence for the logarithmic ratio of the two quantities, ruler length and measurement length. This ratio is summarized as a parameter for the scaling law that is put forth in the book.

The Mystery of Cotton:

This is one of the most fascinating chapters of the book. It deals with connection between fractal geometry and cotton prices. There are three pieces to this connection

  1. Power laws
  2. Power laws in economics
  3. alpha stable distribution

Combining these three aspects, Mandelbrot plots the change in cotton prices vs. frequency count on a log-log scale and finds a remarkable pattern across time scales. He finds that the cotton prices obey an alpha stable distribution with alpha being -1.7. This piece of evidence was stubbornly resisted by many academicians. During the same time period, there was a massive wave of CAPM, MPT, BSM that was sweeping through academia and the industry. Amidst this wave, Mandelbrot’s cotton price analysis did not find the right traction.

Here is the log log plot for up move vs. its frequency, down moves vs. its frequency for Nifty total returns index. image

The slope for the above graphs is approximately -1.1. What do the above graphs say? The frequency of up moves and down moves follows power law. A great many small jumps and small gigantic jumps is what characterizes NIFTY. The above is based on daily returns. One can do a similar analysis at a 5 min interval scale or 15 min interval scale and see that this power law relationship scales across time intervals.

Some people might not believe that such a scaling law exists among financial markets. One simple test that the author recommends is : Take an intraday transaction price data, remove the x axis labels and y axis labels. If you show this graph to someone, they might easily mistake it to be a chart based on closing prices. In physics there is a natural boundary between the laws. The laws that work on macroscopic level are different from that of the laws that work on microscopic level. The scale matters in such field. In finance, there is no reason for such a time scale to exist. Any model you build should take in to consideration this scaling aspect of financial time series.

How does one interpret the value of the slope –1.1 ? It turns out that the slope value for wild randomness is close to -1 ( Cauchy distribution) and slope value for mild randomness is -2 ( coin tossing context). For Nifty, the value is close to –1, i.e wild randomness. Well, one can only wish a dose of good luck to all who are trying to make money in such wild markets.

 

Long Memory, from the Nile to the Marketplace:

image H.E.Hurst

The authors narrate the story of H.E.Hurst , an Englishman, a chief scientist at Cairo who immersed himself in a particular problem, i.e. forecasting the flooding intensity of river Nile. Because the amount of flood varied from year to year, there was a need to create “century storage” to stockpile water against the worst possible draughts. This task was assigned to Hurst. As he began investigating the data, he found that the yearly rainfall was Gaussian. The problem however was with the “runs”. He found the range from highest Nile flood to lowest widened faster than what a coin-tossing rule predicted. The highs were higher, the lows, lower. This meant that not only the size of the floods, but also their precise sequence matter. Hurst went on a data collection exercise and collected all sorts of data without any preconceptions. He looked through any reliable, long-running records he could find that were in any way related to climate, for a total of fifty-one different phenomena, 5915 yearly measurements. In almost all cases, when he plotted the number of years measured against the high-to-low range of each records, he found that the range widened too quickly – just like the Nile. In fact, he found as he looked around the world, it all fit the same neat formula : the range widened not by a square-root law as coin-tossing, but as a three-fourth power. A strange number, but it was , Hurst asserted, a fundamental fact of nature.

This “Nile flood” example is one that is very often quoted to illustrate “Long memory processes”. So, in simple words, these are time series where the dependency dies down very slowly. If you plot the autocorrelation plot, the series dies down slowly as compared to any stationary process. One can easily get the Nile data from the web. This plot shows that yearly flood data is nearly normal but the second plot shows that ACF decays very slowly. The third plot shows a linear relationship between log range and log duration and the slope of the line is estimated as the Hurst coefficient.

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In a chance conversation with a professor at Harvard, Mandelbrot starts working on long memory processes. To paraphrase the author here,

The whole edifice of modern financial theory is, as described earlier, founded on a few simplifying assumptions. It presumes that man is rational and self-interested. Wrong, suggests the experience of the irrational, mob-psychology bubble and burst of the 1990’s. A further assumption: that price variations follow the bell curve. Wrong, suggest the by-now widely accepted research of me and many others since the 1960s. And now the next wobbles: that price variations are what statisticians call i.i.d independently and identically distributed – like the coin game with each toss unaffected by the last. Evidence for short-term dependence has already been mounting. And now comes the increasingly accepted by still confusing evidence of long-term dependence. Some economists, when thinking about long memory, are concerned that it undercuts the efficient market hypothesis that prices fully reflect all relevant information; that the random walk is the best metaphor to describe such markets; and that you cannot beat such an unpredictable market. Well, the efficient market hypothesis is no more than that, a hypothesis. Many a grand theory has died under the onslaught of real data

Mandelbrot incorporates this dependency in his fractal theory and denotes it as H, the Hurst coefficient. For a Brownian motion, H equals ½ and for any other persistent process, H tends to fall between ½ and 1. The more the persistence, the more H will be close to 1. If H is smaller than ½, then the process is in a mean reverting mode. With this context, the authors introduce “Fractional Brownian motion”.

Noah, Joseph and Market Bubbles

The title of the chapter is meant to aid readers to quickly recall the two effects that are present in the financial markets; first characterized by the sudden jumps in the prices and second characterized by the long memory nature of prices. The authors use two stories from Bible and they probably resonate with someone who is familiar with Christian mythology. The takeaway for me is that Mandelbrot’s theory is a combination of two parameters, “alpha” that takes in to consideration the scaling law and Hurst coefficient that tries to characterize the long term memory of the price process. The wildness that characterizes markets can be described in two words – “abrupt change” and “almost-trends”. These are the two basic facts of financial market, the fact than any model must accommodate. They are two aspects of one reality. Mandelbrot also mentions about a test to tease out two effects, a non parametric test called “rescaled range analysis”. But in some cases the two effects are closely related and teasing out the two effects becomes challenging. The chapter also talks about the interplay of Noah and Joseph effect in creating financial crisis at regular intervals.

The Multifractal Nature of Trading time

This chapter combines the two aspects, dependency and discontinuity and mixes it with trading time, a concept explained in this chapter. This combination results in a Multifractal model. Honestly I need to work with this model to get more understanding about the way it works. Pictures do convey a good idea about the model but my understanding as of now is very hazy.

Ten Heresies of Finance

This chapter lists a collection of facts that authors have found useful in their work

  1. Markets are Turbulent – Two characteristics of turbulence are scaling and long-term dependence. These characteristics are found in a number of examples from natural to economic phenomenon
  2. Markets are very, very risky – More risky than the standard theories imagine: Average stock market returns mean nothing to the investor. It is the extremes that matter. That’s where most of the money is made or lost. This is what explains the equity risk premium puzzle.
  3. Market “Timing” Matters Greatly – Big Gains and Losses Concentrate in to small Packets of Time : Ignore your broker’s advise that their clients should buy and hold. Most of the returns(positive or negative) are made in small concentrated time units.
  4. Prices often leap, not glide – That adds to the risk: Continuity is a common assumption and when you backtest a strategy, it is likely that your entry and exit points might seem perfect. The only problem is, in reality, the price jumps and by the time you are in the market based on your entry point, the market has squeezed out a significant proportion of the profit from the proposed strategy. The mathematics of Bachelier, Markowitz, Sharpe and Black-Scholes all assume continuous change from one price point to the next. All the assumptions behind them are false. The capacity for jumps, or discontinuity, is the principal conceptual difference between economics and classical physics. Discontinuity is massively profitable for market makers. You got to model it and incorporate it in your strategy.
  5. In Markets, Time is Flexible – The crux of fractal analysis is that same formulae apply to any scale , be it at a day scale, or an hour scale or a monthly scale. Only the magnitudes differ, not the proportions. In Physics, there is a barrier between subatomic laws of quantum physics and the macroscopic law of mechanics. In finance, there is no such barrier. In finance, the more dramatic the price change, the more the trading time-scale expands. The duller the price chart, the slower runs the market clock
  6. Markets in All Places and Ages Work Alike – This is like throwing away localized market microstructure studies away!
  7. Markets are inherently uncertain and Bubbles are inevitable – This section has a very good analogy, comparing “land of ten thousand lakes with a neighborhood called Foggy bottoms” to “trading world”. Very interesting analogy.
  8. Markets are deceptive – Be aware of the problems with just going with technical analysis.
  9. Forecasting Prices may be perilous, but you can estimate the odds of future volatility – A 10% fall yesterday may well increase the odds of another 10% move today – but provide no advance way of telling whether it will be up or down. This implies correlation vanishes despite of the strong dependence. Large price changes tend to be followed by more large changes, positive or negative. Small changes tend to be followed by more small changes. Volatility clusters. This means that at least you have some handle on it. However you cannot predict anything with precision. Forecasting volatility is like forecasting the weather. You can measure things but you can never be sure enough. This section also mentions about “Index of Market Shocks”, a scale that is used to measure financial crisis.
  10. In Financial Markets, the idea of “Value” has limited value – All value investors will have a shock treatment after reading this section. Value is a slippery concept and one whose usefulness is vastly over-rated

The authors end this chapter with the following quote:

The prime mover in a financial market is not value or price, but price differences, not averaging but arbitraging


In the Lab

The last chapter begins with mentioning academicians, quants and traders who are working on fractals in finance.

imageRichard Olsen

For more than two decades Olsen and his team ( Oanda.com) of mathematicians have been analyzing high frequency data. The authors narrate the team’s history and their work. Olsen started his firm after having a sense of frustration with the way financial models have been modeled and used. The sheer lack of understanding in to financial markets makes him get up every day and head off to do his research. Olsen firmly believes that world is a fractal and hence uses all the techniques from Mandelbrot’s theory to understand and trade in the markets. More specifically, he uses multifractal analysis that rests on the assumption that “trading time” is different from “clock time”.

imageJean-Philippe Bouchad

Capital Fund Management, a hedge fund based out of Europe uses a mix of statistical arbitrage and multi fractal analysis for their trading strategies. The fund firmly believes in scaling by power law and long-term dependence and hence uses a different kind of portfolio management techniques. They call it “generalized efficient frontier” where they focus only on the odds for a crash and their scaling formula minimizes the assets in a portfolio crashing at the same time.

imageEdgar Peters

Edger Peters, another asset manager believes in fractals and has written two books on them. However he does not use them in his funds as he says that his conservative clientele were not interested. In the hindsight of so many crisis that have happened, conservative clientele should have been more than interested in fractals as it comes close to actually explaining the market behavior.

Mandelbrot says that very little work has actually been done in using fractals in finance. He lists the following areas where fractals can truly play a great role

  • Analyzing investments
    • There are so many types of numbers that are seen in the trading/investment world, p/e , volume, EBIDTA,PAT, etc. However when it comes to measuring risk,the industry’s toolkit is surprisingly bare. The two most common tools are volatility and beta of the stock. These two numbers are used again and again everywhere. The assumption behind these numbers is bell curve. How could it possibly be that one and the same probability distribution can describe all and every type of financial asset ? Mandelbrot says that finance today is in a primitive state. Its concepts and tools are limited. The work on scaling, long term memory has just begun. There are many methods to compute Hurst coefficient but no major work has been done show the robustness of methods
  • Building Portfolios
    • CAPM or its variants are all driven by mild randomness assumptions. In 1965, Fama showed that one needs a far greater set of stocks in a portfolio if one makes wild randomness assumption. There is a need for creating a new, correct portfolio theory. Today, building a portfolio by the book is a game of statistics rather than intelligence. In any case, Mandelbrot suggests that every portfolio manager should simulate price series using Monte Carlo methods and test their ideas. Using the methods from the book, one can easily simulate a series based on multifractal geometry and then test out whatever strategy one wants to test out.
  • Valuing options
    • This is another area that cries for guidance. Black schools theory is just that, a theory that makes the very existence of an option redundant. If one can perfectly replicate an option, aren’t options redundant? The observed volatility smile is corrected using a range of fixes such as GARCH, Variance gamma process etc. In the same bank, different groups might be using different ways to value options. Isn’t it strange that we do not have any sensible valuation method for one of the oldest trading instruments?
  • Managing Risk
    • This book was published in 2004 and the author criticizes VAR at length. Somehow no one paid heed to Mandelbrot’s warning. The author is somewhat pleased that Extreme value theory is being used by some people for risk management. However he says that Long memory behavior is not being incorporated in to such theories.

By the end of this book, a reader is thoroughly exposed to the key themes of the book : scaling, power laws, fractality and multifractality.

imageTakeaway:

Most of the financial models taught in universities world over, are “mild-randomness” types. Most of the academic research is concentrated on developing “fixes” to mild-randomness models. However the markets are driven by “wild-randomness”. As long as portfolio managers, traders and quants use “mild-randomness” models, catastrophic losses are inevitable. The book says there are just about 100 odd Fractalists throughout the world who have realized the futility of the current models, have abandoned them for good and are using fractals in their work. For anyone wanting to look at the financial world with fresh eyes, this book is a must read.