Trading Markets have been swamped by a variety of financial assets and subsequently, investment world has witnessed an exponential rise in the number of funds managing these diverse assets. Gone are the days when the investor had to allocate capital between just two segments, bond funds and equity funds. In the current scenario, the options have exploded. The number of Gold funds, sector specific funds, emerging market funds, country specific funds etc have been growing rapidly but the returns across all these investment vehicles have been completely random. Who performs on a consistent basis ? is a million dollar question . In such a situation, what should be investor doing?
This book makes a strong case for a portfolio that the author calls a “Perfect Portfolio”. This portfolio, as per the author, should comprise two Segments. First is a CORE segment which includes Cash, Bond, Domestic Equity and International Equity assets. Second is the Target Market Segment which should have 5 asset classes, Gold, Energy, Agriculture, Real Estate, and Emerging Markets. There is a strong case throughout the book that for each of the two segments, ETFs are the best vehicles as they are
What are the options for an Indian investor ?
In the case of an Indian investor, one can think of building a CORE portfolio using LIQUID ETF like Benchmark Liquid BeES, an equity exposure ETF using Nifty BeES & Junior BeES and International ETFs like Hangseng BeES . Looking at the NIFTY100 returns over the last 5 years, isn’t it hassle free to take a TOTAL MARKET View , rather than spend enormous time in understanding specific stocks. Who in the world knows why a specific stocks moves in the way it moves. Basically if you invest in TOTAL MARKET VIEW, you are using LAW OF LARGE Numbers to your advantage. Look at the NIFTY 100 returns and it appears that TOTAL MARKET VIEW is a safe bet.
If you had invested at the start of the calendar year and redeemed at the end of the calendar year, the NIFTY100 returns are as follows :
||NIFTY 100 Returns
|2010 till date
||79% ( Return from Jan1 to April 26)
The Target Market Segment can be built using Commodity ETFs like Gold ETF . The other asset classes mentioned in the Target Market Segment are not available in the domestic markets for the Indian Investor. But nothing stops him from investing in ETFs in the international markets and create the perfect portfolio.
The following stats from goldprice.org shows the importance of having an alternate asset in the portfolio where there is an obvious benefit of diversification.
Indian investors reaching out for an energy ETF traded in the foreign markets for their portfolio composition is doubtful. If and when such ETFs trade in Indian markets, then I am certain that retail investors will be more receptive to such investments .
This book has one big flaw though. It talks about market timing, trading signals and trading strategies associated with 9 asset classes. Now that is quite a stretch. You can’t expect a retail investor to keep automated trading rules, conducting TA, and take active buy and call decisions and that too with instruments like ETFs which are basically ZEN instruments, meaning, BUY and be at peace with the volatility associated with them. Holding such ZEN instruments and actively trading them as a side activity is a tough thing to expect from retail investors.
The broad message of this book is extremely clear. Diversified ETFs with broad asset classes is the PERFECT PORTFOLIO for a retail investor.
There are two issues which have been occupying my mind ever since the trades that I had suggested have gone wrong.
Type I error( trade when there is no signal) , i.e trade when the series is non stationary. Typically this is relevant to unit root testing where the null hypothesis says the series is random.
Is the spread really a result of cointegrated time series ?
Each issue requires detailed understanding of time series concepts. This post deals with the first aspect.
Unit root testing is one of the most involved areas of econometric analysis. This test requires a working knowledge of probability distributions , stochastic calculus, matrix algebra , Monte carlo simulation techniques to name a few. Needless to say , there are a lot of techniques for unit root testing and the literature is extensive. However the literature is sharply divided in 2 categories. The first category comprises explanations which are intuitive and that is all there is to it. You can probably explain and verbalize the intuition behind it but as a practitioner you would have difficulty in actually testing/ carrying out unit root tests. The second category comprises dense econometric literature where there is extensive use of stochastic calculus, measure theory and simulations. So, there is an obvious need to bridge this gap in the literature and the book , “A Primer for Unit Root Testing”, serves exactly this purpose
This is probably the first time that I am writing about a book which is not even released . Amazon Store says that the book is slated to be released on April 27th 2010. The book is insanely priced at $110 , a tag which is too high for a book which claims to be a primer!. Anyways , the book is available as a torrent obviously at $0 🙂
There can be no long-short trade if you are not reasonably certain that spread is a sample realization of a stationary process. So, in that sense, stationarity property is crucial for statistical arbitrage trading. In stats jargon you need to be sure that the series does not have a unit root. Actually the terminology arises from the fact that the roots of the characteristic equation should lie outside a unit circle and hence the name called the unit root, Alternatively, the eigen vectors associated should all lie with in a unit circle.
This book is titled Primer to Unit root testing and the core of unit root is actually covered in the last chapter of the book. There are about 8 chapters in the book and the last chapter is on unit root tests! Then what is contained in the first 7 chapters. All these chapters deal with the principles that would be needed to do a basic unit root test. Well to do an unit root test, it is easy to run a command in R/Matlab . But if you want to understand the nuts and bolts of various unit root tests and structural tests, a solid understanding of stochastic processes is a must and here is where the book fills the need. For those who have gone through Shreve, the first 7 chapters will serve a quick recap/ reminder of stuff that is present in Shreve.
Chapter – 1 :Introduction to Probability and Random Variables
The first chapter , “Introduction to Probability and Random Variables” , provides a basic introduction to measure theory, probability spaces, random variables, stationary processes etc. These are the basic principles based on which stationarity conditions can be tested.
Often the math behind the symbols Ω, F , Ρ , Β in many books is explained in such a detailed manner with symbols, theorems, proofs etc that the basic intuition behind these symbols is lost. Pick any book on probability, the measure theoretic introduction is so abstract that it requires tremendous amount of motivation to get through the initial stuff. This chapter gives an intuitive feel of Sample space, Sigma fields, Borel Sets, Borel functions, Probability functions etc. The important point of defining probability measure for a Borel set and Borel field in R is intuitively explained by the issue that it is painful to define probabilities for points. Probability measures are assigned to Borel sets in R and hence there should be tools to define and manipulate these measures. These tools are nothing but the cumulative density function and density functions with respect to the random variable.
The connection between (sample space Ω, F sigma field of an Algebra, P probability measure) and (Real Space , Borel sets) in the real space is necessary to understand the basic principles of stochastic process. One must intuitively understand the connection between (Ω, ƒ , Ρ) ↔ (R,Β) and then know the math behind it. Often times, especially students who learn this in a formal course at a university, find that the equation is reversed. They will know the definitions and proofs of things relating to these but do not intuitively understand the stuff. Obviously, they will never be able to apply to practical problems.
There is also a mention of basic definitions of conditional probability, function of random variable, independence of random variables, expectations, covariance structure, correlations, Basic laws of expectations relating to random variables etc
A mention of law of iterated expectations in this chapter is made, intention being that it will be useful in understanding calculations relating to stochastic processes and unit root testing. What is law of iterated expectation? Taleb puts it in a nice way, in his book, “Blackswan”
If you expect that you will know tomorrow with certainty that your boyfriend has been cheating on you all this time, then you know today with certainty that your boyfriend is cheating on you and will take action today, say, by grabbing a pair of scissors and angrily cutting all his Ferragamo ties in half. You won’t tell yourself, This is what I will figure out tomorrow, but today is different so I will ignore the information and have a pleasant dinner. This point can be generalized to all forms of knowledge. There is actually a law in statistics called the law of iterated expectations, which I outline here in its strong form: if I expect to expect something at some date in the future, then I already expect that something at present.
The chapter then introduces stationarity. Strong stationarity arises when the joint distribution of the random variables is time invariant. This is far more difficult to test than weak form. Weak Stationarity arises when the stochastic process has a constant mean, constant variance and a covariance structure that is time invariant. Usually strong stationarity means weak stationarity but it is not always the case. There are examples where one cannot calculate moments and hence a series which is strongly stationary need not be weakly stationary. Partial sums of variables also give rise to weak stationary series which are extensively used in unit root testing.
Prima facie, the importance of stationarity lies in the fact that process parameters can be estimated by choosing any sample portion of the series. It does not matter which sample portion of the data is used for parameter estimation. One important aspect to be kept in mind is that, stationarity refers to a property of the process generating the outcomes. So, one must use the terms stationary process or non stationary process instead of stationary data / non stationary data. The word stationary data has no meaning. Stationarity is a property of the process. But I guess it is ok to use the words in whatever manner one wants to, as long as the intent is clearly communicated.
Chapter – 2 :Time Series
This chapter is a very basic intro to time series through lag operators. The usage of lag operators in simplifying calculations relating to arma processes are shown. White Noise, IIDs, Normal IIDs , Martingale Difference Sequences are mentioned as these form the components of ARMA processes, based on the kind of model that one is dealing with. Most of anti-quant group might object saying, “How the hell do you know that error terms follow some DGP( Data generative process) ? “ for which nobody has an answer. In any case if you want to do something with the data, than just be in the zen state , there one can use time series fundas and atleast try to get some hang of the data patterns. Whenever someone asks this question, my only answer would be “ there is no harm in modeling when you are dealing or living in the HIGH FREQ WORLD“.
Invertibility is discussed in the context of computational necessity for estimating the parameters. Measure of persistence is defined and the first glimpse of the ways to measure non stationarity is provided. AR polynomial equation which has a unit root or roots with in a unit circle behaves in such a way that measure of persistence does not converge and increases with out limit. A whole lot of ARMA estimation procedures are covered at a 10,000 ft level often referring to the estimation, inference of parameters to some software results. One of the takeaways from this chapter is the concept of long term variance. In most of the books, the sigma equilibrium or Gamma0 is mentioned as equilibrium variance. This point about long term variance and the importance of ACVF for various lags in the calculation of long term variance makes a lot of sense, which is something that is not mentioned in a lot of econometric books!
Chapter – 3 : Dependence and Related Concepts
If you are given a sequence of prices, then the first question that a trader might be inclined to ask is about the memory of the process. Is it a weak memory process or a Strong memory process? By weak memory , one means that whether the series of acvfs are absolutely summable and they converge . Long memory obviously means that the absolutely summability does not converge to a specific number but diverges.
The concept of strong mixing is introduced in this chapter , which is not often mentioned in econometrics books like Tsay, Shumway and Stoffer( just one place where the term is mentioned in the context of spectral analysis),Jonathan & Kung,etc. This is relevant for two reasons, one for invoking CLT for stochastic sequences and second , the invariance principle which is useful for unit testing.
Basic definitions of ergodicity is defined with a nice illustration which brings out the difference between ensemble averages and time averages and the need to invoke asymptotics to understand the properties from the realization of a Random variable. Martingale and Markov properties of random variables are also mentioned so as to lay the foundation for understanding the tools of unit root testing. The chapter ends with a description and properties of a poisson process which is a stochastic process with Markov property.
Chapter – 4 : Concepts of Convergence
This chapter is primarily about convergence and orders of convergence. Convergence is a very important concept which is very clearly explained in this chapter. Convergence in Distribution, Convergence in Probability, Almost sure convergence, Mean Square convergence and related concepts are clearly explained with examples. Besides, the type of convergence , the orders of convergence is also important in choosing between estimators. Big O and little o notation are introduced in the context of stochastic processes. Finally convergence of one stochastic process to another process is discussed though in a non technical manner.
One thing I would like to mention related to this context. You pick up any book, the # of pages devoted to modes of convergence is not more than 20 pages in any standard text book. Somehow the authors assume that readers/students get it once they are through with 20 pages. One quick way to empirically test this is : Take a random sample of math fin students graduating out of top 10 univs in US. Ask them to code a function to check for almost sure convergence , convergence in probability and convergence in distribution. The % of students who would successfully be able to code would be extremely low, if I can extrapolate from whatever I have seen during my masters. You would even come across people who would say that they would have read Shreve cover to cover but would stutter when asked the key differences between modes of convergence. If one thinks about it, modes of convergence is probably one of the most important aspects of math finance and it is given abysmally low coverage in most of the places. What’s the remedy ? Well, I think someone should create a bootstrapped version of testing all modes of convergence and atleast give the students a sense of visually seeing the various modes of convergence. I firmly believe that unless you visually see something, your learning would be half-baked and in that belief I sincerely hope that there would be a good data visualization tool someday so that students can appreciate the modes of convergence. Anyway that was an unnecessary digression from the intent of this long post.
Chapter – 5 : Introduction to Random Walk
When it comes to random walks the law of large number reasoning falls apart. If you simulate a large no of coin tosses and create a partial sum of series which is basically an arithmetic random walk, the statements you can make about the some events would surprise you . Here are a few things which are not intuitive
If you create an 2N step Arithmetic random walk where the outcomes are of a binomial process, and compute the last time at which the sum of heads and tails was 0, you might assume that the value would be around N. However a quick simulation shows that it is an arc sine distribution.
Similarly the fraction of time the walk spends on the positive quadrant, again follows an arc sine distribution. It is not at all 50:50, ( this is usually present in most of the books as gambler’s ruin problem).
Sign changes is another thing where the simulation show a completely different result. Let’s say you want to find out the average sign changes of a random walk which contains 100 steps.. CLT knowledge might make you guess some number but simulation shows that it is close to 3. So, these are some of examples where your usual notion of intuitive feel will not be of help. This chapter scratches the surface of random walks, but I feel that it is clear enough to make you dive in to any book on stochastic calculus.
Chapter – 6 : Brownian Motion
Brownian Bridge was a topic which I was eagerly looking forward to, in my masters program. My faculty who was a PhD from Courant, for some reason did not cover it, even though Shreve had it explicitly mentioned. While I was attending the course on Stochastic Calculus, I was also working on Paul Glasserman’s book. It was clearly mentioned in Paul Glasserman’s fantastic volume on Monte carlo that Brownian Bridge is a method for variance reduction while simulating random paths. That was the time when I was just falling in love with simulation techniques. Immediately I went and asked my faculty whether she would be covering Brownian bridge in her class. She promptly said ,”No, you read it yourself”. Then she went on and on about how pure mathematics was being ruined by fraud applications to finance etc etc. I don’t know what she went through in her professional life. But I was surprised that a PhD from a reputed institute had no energy or enthusiasm towards finance , a program where she was a faculty!!. Well, I went back and spent some time understanding it .This incident happened in Jan 2008.
After 2 years, I came across Brownian Bridge again as I had to use in my work. This time again, I came to understand that Brownian Bridge is used heavily in Unit root testing, something which every arb trader needs to know. This time I was resolute and wanted to understand it thoroughly and not merely knowing it from 10,000 ft.
Personally, Visualization is the my only way to learn stuff. So, simulated data and started working on Brownian Bridge.
To begin with, you can see what a SBM(Standard Brownian Motion is)
If you take a symmetrical random walk and scale the increments so that they are dependent on the time step, basically you get a Brownian motion. This can be easily checked by normalizing the series and validating against standard normal. The above graphs show that as you increase the time steps for a given T, scaled random walk divided by the variation, converges in distribution to standard normal.
Basic properties of BM are introduced where there is a special mention of quadratic variation converges to T.
The whole ram kahani of Brownian Motion and Brownian Bridge leads to the concept of Functional of Brownian Motion which is used heavily in Unit root testing. So, if you are a math fin student trying to read stochastic calculus, my humbe suggestion is not to miss “Brownian Bridge” at any cost as it is used in a lot of applications.
Chapter 7 : Brownian Motion Differentiation and Integration
If you compress Stochastic Calculus by Shreve in to 30 odd pages removing most of the math and present it in a completely intuitive sense, then you basically have this chapter in print. The chapter highlight the limitations of Reimann Calculus and shows that Brownian Motion calculus is a different animal where everything is a Lebesgue integral. So, in that sense this is quick recap of Ito’s calculus, Brownian Bridge , etc which are all very important to understand any unit root test , starting from the basic Dickey Fuller.
FINALLY , the chapter on Unit Roots
Chapter 8 : Some examples of Unit Root Tests
This is the core of the book even though this is last chapter of the book. As mentioned earlier, the first 7 chapters is to prepare a reader to understand the material in this chapter.
There are basically two statistics that are typically used. Normalized bias and Pseudo t statistic. Normalized bias is powerful than pseudo t statistic but the latter is much better than the former whenever the error process has a correlated structure
The usage of Functional Central Limit Theorem is something any first timer would notice. The connection between Brownian Bridge and pseudo t statistic will be a revelation for people who just use some black box to test stationarity of the series / who do a quantile plot to show stationarity or the lack of it. Basically depending on whether you are testing a series which reverts to 0 or reverts to a constant or has a linear trend, the limiting distribution converges in distribution to various forms of Brownian Bridge. Most of them do not have closed forms and hence simulation procedures are resorted to understand the stationarity aspect.
All the p values for the t statistics are available in the paper by Dickey ( 1979). However the basic form of Dickey ( 1979) does not address lags. Hence augmented Dickey Fuller test is used. MacKinnon tables are also used .. But one thing to keep in mind is that , all these tests are basically unit root tests where the null is non stationary and alternate is stationary. So, if you are basically trying to trade a spread, it is better to establish stationarity using KPSS type of test where the null is stationary and alternate is non stationary. Thus you are using two tests , firstly the unit root test which has null : Non stationary , alternate: stationary.. secondly , KPSS where null = stationary ; alternate = non stationary. So, your test should show that you reject null in unit root test and accept null in KPSS test ..Only then you have something that is tradable.
Elliott–Rothenberg–Stock Test for Unit Roots is also mentioned in a very detailed manner which basically improve the power of unit root tests.
Personally the non parametric test mentioned at the fag end of the book has been a great learning. The idea is straightforward but the math needs to be understood carefully. The basic idea is this :
If you have a random walk with out drift and you estimate the test statistic relating to level crossing times for a specific (s), it converges in distribution to a folded standard normal distribution. If the random walk has a drift, the test statistic relating to the level crossing times converges in distribution to Rayleigh Distribution. If the error has a correlation structure , one needs to detrend and the test statistic again converges in distribution to Rayleigh. I will definitely this soon as I am particularly inclined to use semi-parametric type of statistics than using only parametric stats.
This book is a great bridge between “having an intuitive notion of unit root” and “having an in depth knowledge of various unit root tests that you can create your own unit root test ”.
The title of the book makes it obvious for anyone what’s in store for them in the book. At some point of time or the other, we all would have made a list of something in our lives… Grocery list, Books to read list, Things to do list, Steps in an algo while coding, List of aspects to be checked in unit testing/system testing, List of steps in a process (that is outsourced to a call center where standardization is the key) , List of items to be back packed, etc.
Personally I like making a checklist even if the list is not more than 4 items long. The very fact that you tick off the item after you complete a task gives me some kind of feel good effect that you have accomplished whatever you had intended to. In one way it also quantifies things that you have in your mind. those 6 books that you wanted to read, those 5 things you need to have while taking a customer call , those 9 items you should buy at a grocery store etc .. In that sense a checklist is a nice and simple quantification mechanism, a recall mechanism for our attention strapped life.
Ok, now is there a book worth of stuff relating to checklists ? If the would have been written by a management consultant, or a self help guru, I wouldn’t have bothered to even cast a glance at it . But the fact that it is written by a doctor who talks about the usage in the medical field is something that caught my attention. Medical field at the outset seems like a field where there is no scope for standardization as every patient needs to be treated differently. However the author says checklist can get things right in medical field..now that should any reader some one curious to see how a simple check list can be useful.
The book starts off with a few examples of doctors who have gained enormous experience in surgery, became highly skilled over their careers , but still fail to do things in an organized way while performing surgeries. So, what’s missing ?
The author brings out a fundamental puzzle in modern medical world. Because of sophisticated treatment procedures, a doctor needs to keep track of innumerable tasks and more often than not , he/she adopts specialization to super specialization path. This solution has not been quite effective as there are about 150,000 deaths following surgery in US every year, more than 3 times the road casualties.
The book takes a tour of various domains where checklists have been used , sometimes by accident, sometimes by necessity . Though, the form , format and content of checklists have evolved in these domains, the tool has made a lasting impact. Basically if you look at all the places where checklists are used, they started as a means to overcome fallibility of human memory and attention , when it comes to mundane and routine matters. Slowly the checklists were adapted to the situation and communication mechanism was put in place to make it work. In essence, any system driven problem solving might need a checklist in its tool kit.
In the construction industry where the very activity of project involves a multitude of tasks at varying complexity levels, there is a heavy use of checklists in various forms, routine task list, exception list, etc. An industry which was characterized by compartmentalized work culture of design-implementation has been replaced by more wisdom of crowds thinking .A single overarching design is not meeting the trying demands of a multi-storied structure. Construction industry has thus seen an end to the master builder culture and has taken a systems approach to design + implementation.
In the airlines industry, checklists are heavily used to handle various situations and remove mental hijack when there is a panic. By laying down the procedures for emergencies, the airlines makes sure that crew does not forget to do some of the basic steps of an emergency. Did you know that the pilot for the airbus that averted the accident, which was called miracle on Hudson and was covered in the media for days and weeks , had used checklists and his crew were well trained to handle such situations. Even though the media portrayed it as an incident where a single pilot had a great role to play, a little investigation by the author revealed a different story, LUCK + Preparedness(checklists) was behind the Miracle on Hudson.
In the field of hotel management, one of the finest kitchens in the world use and communicate using checklists. Sometimes the staff go to an extent of keeping a checklist for each of its valuable customer to give the same consistent experience time and again.
There are a few examples of investors , venture capitalists who use checklists in their work to finalize a deal. There is a mention of Pabrai Investment Funds California, Acquamarine capital management in Zurich who diligently use checklists. Now one can dismiss them as freak examples.
So, if there are so many areas where checklists are used, why hasn’t been an adoption amongst various fields/people ? In the words of the author,
“We don’t like checklists. They can be painstaking. They are not much fun. But I don’t think the issue here is mere laziness. There’s something deeper, more visceral going on when people walk away from not only saving lives but from making money. It somehow feels beneath us to use a checklist, an embarrassment. It runs counter to deeply help beliefs about how the truly great amongst us – those we aspire to be – handle situations of high stakes and complexity. The truly great are daring. They improvise. They do not have protocols and checklists. May be our ideas of heroism needs updating
The fear people have about the idea of adherence to protocol is rigidity. They imagine mindless automatons, heads down in a checklist, incapable of looking out of their windshield and coping with the real world in front of them. But what you find when a checklist is well made, is exactly the opposite. The checklist gets the dumb stuff out of the way , the routines your brain shouldn’t have to occupy itself with.”
Making a checklist and adhering to it enforces discipline, and reduced human fallibility. I think it is a good quantification mechanism where you can see what’s done, what needs to be done and where you stand.