Exploring Operational Definitions: Part I

An operational definition is a way to define a concept by the set of operations or procedures which are used to make a judgement with that concept. This idea, along with a philosophical thesis on meaning known as operationalism, was popularized by Percey Bridgman (see his The Logic of Modern Physics).

One example he liked to use is the concept of distance.

More on Benjamin Graham and Uncertainty

So in Uncertainty and the Margin of Safety, I suggested that an important concept in physics ought to be applied to investment analysis (and economics for that matter). Furthermore, I suggested that Benjamin Graham's concept of "margin of safety" was linked with this idea of uncertainty.

Today I want to take a closer look at the sorts of uncertainties faced in investment analysis and how Benjamin Graham recommend one face those uncertainties.

Stock Valuation and Anchoring: AF2P Contest Results

Thanks to everyone that participated in my little study. I'm going to outline what questions I wanted to answer with this study. Hopefully, you will find this information useful.

Introduction

As investors, we're trying to find good companies trading at a good price. But how do we assess what to pay? Obviously there is a good deal we don't know; we must make decisions under conditions of uncertainty.

Enter the AF2P Stock Valuation Contest!

So we're having a little stock valuation contest. It shouldn't take you too long and you have a chance to win \$25. You'll be helping me out in a little research project I have so I'd appreciate it if you would sign up and participate.

Click here for details:

AF2P Stock Valuation Contest

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Thoughts On Backtesting

So I have a few methodological thoughts on backtesting strategies. The lack of sound methods in studies is problematic in my opinion. If anyone has some insight into this I'd appreciate it.

I'm going to lay out what I consider three common problems in empirical research into backtesting investing strategies. Not all studies suffer from all three problems but I think a lot do. And I think attempting to address these is a good start.

Arithmetic and Geometric Returns

Many times when expressing returns, the arithmetic returns are used instead of geometric returns. This is actually quite problematic. But there are ways of actually relating the two returns which I'll share today.

A Unique Solution to the Monty Hall Problem

The Monty Hall Problem is a counterintuitive result in statistics. The problem goes something like this.

Sketch of the Monty Hall Problem

Suppose there are three doors: A, B and C. Behind one of the doors is a big prize (a car, a huge pile of cash). Behind the other two doors are just some goats.

Now you make a selection of a door but the game is not over with yet. Suppose you pick door letter A. Now the game is not over with yet at this time. The host actually reveals one of the remaining doors, say door letter C, to have a goat.

Now the host gives you the option: stick with door A or switch to door B.

Thoughts on the Sharpe Ratio

So this is going to be just a few musings on the Sharpe Ratio. But before that, I want to do a comparison to a technique I utilized because it has similiarities to the Sharpe Ratio. This is also somewhat related to Cullen Roche's question and my response here.

In my blog post, Are Bond Yield Spreads Adequate?, and the subsequent follow-up, Junk Bonds: A Closer Look, I developed a simple model to analyze spreads to see if they were adequate. Today I'll give a more "intuitive" explanation of that model.

Some Thoughts on Risk/Return Tradeoff and EMH

So this is partly a response to a question asked by Cullen Roche on Twitter:

@dvegadtime @financequant Q: What's your preferred measure of risk adjusted return? Sharpe, SDR Sharpe or Sortino? Thanks.
— Cullen Roche (@cullenroche) October 19, 2013

Is the Stock Market a Ponzi Scheme?

Today I want to explore the question on whether or not the Stock Market is a Ponzi scheme. The reason why is that I think many people view it as such but may not even realize it. So the big question here is this: are they right?

Charles Ponzi's Scheme

On [Logical] Equivalence

In Newton's Law is F=ma?, explored the issue of whether or not this formulation was equivalent to Newton's actual statement of his 2nd Law. Today I want to further explore the topic of [logical?] equivalence. In general, I want to know what it means to say that two statements, $P$ and $Q$, are equivalent.

Junk Bonds: A Closer Look

So one of the questions I asked in Are Bond Yield Spreads Adequate? is whether or not junk bond spreads are adequate. I presented a simple model. The model predicted that returns on junk bonds would be about 2.27% in excess of treasuries. But the uncertainty in the model had a standard deviation of 2.36%. So if we were off by just 1 standard deviation, we would underperform treasuries.

But a model can't be better than the assumptions that one puts into it. I'd like to take a review of the assumptions I used in the model and change a few things. 

Financial Mathematics: Perpetuities

So we're continuing our look at annuities. A perpetuity is an annuity that continues to make payments indefinitely or perpetually, hence the name perpetuity.

For simplicity, we shall make our usual assumptions:

1. All payments are made in equal amounts.
2. The payments are made at equal intervals (specified as a fraction of a year).
3. There is no possibility to alter either the times or the amounts of the payments (e.g. no prepayments).

Newton's Law is F=ma?

This is a brief exploration on the logical structure of Newtonian mechanics. 

When physicists teach classical mechanics, they often refer to Newton's 2nd Law as "F=ma" or more technically as:

Growth at 2% Per Year

So one thing that frequently occurs is that we assume growth will continue at a constant rate. I mention 2% because this is often seen as the long-term real growth rate of an economy. What I want to do is get some perspective on that.

Let's suppose we're living 6000 years ago, circa the 4th century BCE. If I started with 1 of something (it doesn't matter at this point what it is) 6000 years ago, and it grew at 2% per year, how much would I have today? The answer is simple:

Financial mathematics: Annuity - Geometric Progression

So we'll look at an annuity that has a geometric progression.

Suppose that you want payments every year but instead of each payment being the same, you want to be some multiple of the previous payment. For example, you may want the payments to increase every year at $ 3\%$ to keep up with inflation. So we're going to introduce a growth term into the formula.

Are Bond Yield Spreads Adequate?

Suppose you have the choice between three classes of assets: treasury bonds, investment grade corporate bonds and speculative grade corporate bonds (junk bonds). Which one should you choose?

Many people wrongly just look at yields. If you look at yields, the answer is simple: choose junk bonds. Junk bonds offer a higher yield.

The problem is that doesn't account for the fact that some junk bonds default and the losses that result from that.

The second issue is the uncertainty in modeling losses. I'm going to be using some models that require assumptions. These assumptions are not perfectly known. So we need to build in a margin of safety to make sure that we outperform treasuries.

The Structure of Arguments: Deduction, Induction and Abduction

So this is a brief overview of three types of logical arguments.

My 104% Per Year Investment!

OK, so some of you may recall my blog entitled My 1348% Per Year Investment and are thinking, is this going to be another one of those?

And the short answer to that is "yes". But there will be a bit more to this. As this is a quasi-promotion for a quasi-textbook that I (a quasi-person) am writing. That text is on Financial Mathematics. You can also find it via the huge banner at the top.

Financial Mathematics - Loans - Annuity Due

We're looking at different types of annuities. In the previous section, we took a look at annuity immediate loans. In this section we'll be dealing with annuity due loans.

 Annuity Due Loans

Bond Returns Given a Change in Interest Rates

This post was inspired by a tweet by John Hussman:

Bond Asymmetry 101: At 2.9% yield, 50 bp increase in 10-year benchmark Treasury yield => -1% total return over 1 year; 50 bp decrease => +7%
— John P. Hussman (@hussmanjp) September 6, 2013

Financial Mathematics: Loans - Annuity Immediate

We have already started looking at annuities. Today we'll be continuing that discussion. We'll be using two common examples to illustrate annuities that most people are familiar with: Loans and Savings.

This section will be dealing with loans and particularly the annuity immediate loans. In the following sections we will explore annuity due loans as well as savings formulas.

P/E to VIX Ratio - Are you complacent or skeptical?

So Bloomberg had a chart of the day from David Bianco relating the Price to Earnings ratio for the S&P 500 and the VIX (which measures implied volatility of S&P 500 options). Bianco used this as an indicator of sentiment, whether investors are complacent or skeptical.

I took a look at this using data from Robert Shiller and yahoo finance:

Relating ROE with ROA and Leverage

This is just a quick derivation where I use some simplifying assumptions. I'm going to show that return on equity (ROE) is a function of return on assets (ROA) and leverage. This post was partly motivated by a discussion here.

Here are the assumptions:

Useless Stock Metrics

So today I'm going to discuss metrics used in stock analysis that I think are useless and largely uninformative. In spite of this, many of these are popular and I would like to suggest they shouldn't be popular. (Granted, I think "pop" music shouldn't be popular so what do I know?)

Some of this will be an extension of a conversation at OSV forum (Firm Versus Equity (apples with apples)) as well as a great blog post by Prof Damodaran on the same subject (A tangled web of values: Enterprise value, Firm Value and Market Cap ). 

So without further ado . . .

Financial Mathematics: Flow of Funds Table

A Flow of Funds Table is a simple heuristic that can be useful in understanding assets, especially in more complicated scenarios.

The flow of funds chart has two parties which I call Buyer and Seller. Initially, the Buyer exchanges Capital for an asset; the Seller exchanges an asset for Capital. Eventually at some settlement period (which may be multiple periods), the Seller will make payments back to the buyer.

Shareholder Yield (Quasi-Book Review)

Mebane Faber has a nice read entitled Shareholder Yield: A Better Approach to Dividend Investing. If you keep an eye out, you may be able to get it the kindle edition for free. Regardless, it's still less than $6 for either the Kindle or paperback edition. This will be a quasi review/discussion of the book.

To be successful, managers of a company need to be good at two things: operations and capital allocation. While Faber notes many books focus on operations, the focus of this book is devoted to (a subset of) capital allocation.

Capital allocation concerns itself with whether or not to obtain financing, what type of financing (debt, equity, preferred, etc), how and when it should be employed, and how and when it should be paid back. Faber's book is concerned with the latter aspect of paying back financing.

Financial Mathematics: Geometric Series

For a refresher on sequences and series, see here.

A geometric sequence is a sequence in which the following term is a multiple of the previous term. For example:

What about the Fed's Balance Sheet?

So I often see a lot of discussion regarding the Fed's Balance Sheet related both to its size and to its liabilities. A recent example of a question by John Hussman:

Should Fed be subject to capital requirements? A $3.7 trillion balance sheet and $55bn of capital is 67-to-1 leverage http://t.co/OuX3KFLpng
— John P. Hussman (@hussmanjp) August 16, 2013

The main question I think to ask is: Why does the Fed's balance sheet matter?

The corollary question is this: In what sense is the Fed's balance sheet like that of a balance sheet of a private bank or nonfinancial company?

On why #themarketis up (down) today

So I'm suggesting that a new hashtag be used: #themarketis. The idea is to use it to explain why the market is up (down, etc) on a given day. Why, you ask? Allow me to explain.

Almost every day there's a "news" story on the market's price movements. Many of them offer explanations (rationalizations) on why the market price has moved. The explanations sound plausible but I think in most cases they are, at best, simplistic hypotheses which would not stand up to a robust statistical analysis.

Keynes on Investment, Speculation and Uncertainty Part II

In the first part of this series (duo?) we began looking at Chapter 12 from John Maynard Keynes' General Theory. Today we will finish that discussion a bit.

In the first part we looked at the role uncertainty has in investment. In particular, Keynes notes that there is great difficulty in predicting future variables regarding investment opportunities. As a result we mainly rely on the convention that the future will, more or less, behave like the recent past.

Financial Mathematics: Sequences and Series

In mathematics, a sequence is an ordered list of numbers. A series is the sum of the terms of a sequence. Series are also a kind of sequence. Series are an essential tool in dealing with certain kinds of financial instruments.

Financial Mathematics: Annuities

An annuity is any stream of payments.  Examples include savings accounts where regular deposits are made, loans in which regular payments are made, annuities (the financial product offered by many insurance companies) and so on.

There are a variety of types of annuities and it would be difficult to cover them all. We'll look at a few common ones to get a flavor for how annuities work.

Keynes on Investment, Speculation and Uncertainty Part I

John Maynard Keynes is, as an economist, either well-liked or greatly despised. I suspect a lot of that dislike is a result of the neoclassical synthesis which had more to do with Hicks and classical economics than it did with Keynes but all of that are just mere footnotes.

Keynes' most well known contribution is The General Theory of Employment, Interest and Money or sometimes just referred to as The General Theory.

If you've ever tried to read The General Theory you probably noticed it's not exactly an easy book to read. Part of this I think has to do with paradigm shifting. Often people who are rethinking the way we think about a particular discipline or subject have a hard time conveying their ideas. They either have to invent new terminology (which they may do a poor job of showing how to use that terminology) or they use old terminology in new ways (which comes with the baggage of the old ways of using that terminology.) Later thinkers then have to attempt to interpret what's actually saying and apply it which is often no easy task. But I digress.

Follow-up On Investment Poll

In an earlier blog I presented a poll with three alternatives (if you haven't taken the poll, see here). There's some observations at the above blog as well as at Fool and OSV.

My 1348% Per Year Investment!

So several years ago I made a very small investment which has returned quite well. I made my money back within a few months and the rest was pure profit.

Altogether, I estimate that my returns have been 1348% per annum.

Financial Mathematics: Financial Calculators

Actually working out most of these formulas by hand can be time consuming. In some cases, there is not a slick algebraic solution in which case you'll have to use brute force calculations or some approximation technique (to find the interest rate). So here's a quick review of some alternative ways do actually do the calculations.

Financial Mathematics: The Rule of 72

At some time or another you've probably heard of the "rule of 72". I'm going to give a derivation of this result and show how good of an approximation it is.

On GMO's 7-Year Asset Class Return Model

Every month GMO gives forecasts for several different asset classes (sign up for free to access these). What I'm going to do a quick look at is the model they use to derive these projections.

Poll: Which of these investment options do you prefer?

So the following is a poll I'd like to conduct. Feel free to post thoughts on your reasoning. I'll present three investment alternatives. Your job is to order them by preference.

Financial Mathematics: Continuous Compounding Interest

Earlier we set up the basic compounding (discounting) factor for when interest is compounding at smaller intervals than 1 year:
$$\left(1+\frac{i}{m}\right)^{nt}$$
where $i$ is the interest rate, $n$ is the number of divisions in a year and $t$ is time expressed in years.

Next we're going to suppose that compounding occurs, instead of annually, quarterly, monthly or even daily, but rather in a continuous manner. This means that our time interval goes to 0 or the number of intervals in the year is infinite.

Financial Mathematics: Periodic factor, APR and APY

So in the first piece on the Time Value of Money, we looked at scenarios in which interest was compounded annually. But interest is not always compounded annually. Some assets compound semiannually, quarterly, monthly or even daily. So we need to develop some tools to deal with these.

To begin with we will introduce some terminology.

Financial Mathematics: Time Value of Money

One of the critical ideas that drives much in financial mathematics is known as the Time Value of Money (TVM). So I think it's important to start here. In fact, if you do not understand TVM, you will not understand the remainder of this series as TVM is an implicit assumption in most of these models.

The key idea behind TVM is that a dollar today is not worth the same as a dollar tomorrow. Part of what motivates this idea is postulate about human behavior.¹ That idea is that human beings prefer to have something (money) now rather than later. As a result, money in the future is worth less today than it is in the future.

Financial Mathematics: Table of Contents

This is the beginning of a series of blogs on financial mathematics. The purpose will be to motivate the techniques used as well as derive explicit results. The presentation will be for those interested in seeing the logic behind the equations and what assumptions are made to derive them.

Feel free to bookmark this page as I update content.

You are more than welcome to post feedback on the series. My goal is to explain the concepts in a way that promotes understanding. If there are any errors, confusing concepts or anything else that needs clarifying, feel free to let me know.

Table of Contents

1. Time Value of Money
1. Periodic Factor, APR and APY
   
2. Continuous Compounding Interest
   
1. The Rule of 72
2. Annuities
1. Loans - Annuity Immediate
2. Loans - Annuity Due
3. Geometric Progression
4. Perpetuities 
3. Bond Valuation
4. Stock Valuation

Appendix

1. Financial Calculators
2. Sequences and Series
3. Geometric Series  
4. Flow of Funds Table
5. Statistics - Expected Values  
6. Statistics - Moments

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Poincare on Two Types of Mathematicians

In Henri Poincaré's book, Science and Method, there is an interesting discussion on mathematics education. I'll quote a relevant portion:

Many children are incapable of becoming mathematicians who must none the less be taught mathematics; and mathematicians themselves are not all cast in the same mould. We have only to read their works to distinguish among them two kinds of minds - logicians like Weierstrass, for instance, and intuitionists like Riemann. There is the same difference among our students. Some prefer to treat their problems "by analysis," as they say, others "by geometry". 

It is quite useless to seek to change anything in this, and besides, it would not be desirable. It is well that there should be logicians and that there should be intuitionists. Who would venture to say whether he would prefer that Weierstrass had never written or that there had never been a Riemann? And so we must resign ourselves to the diversity of minds, or rather we must be glad of it.

Uncertainty and Margin of Safety

A topic that comes up frequently, and one that I think is not well understood, is the concept of "margin of safety" in value investing. The idea goes back to Benjamin Graham. But before we do that I want to take a detour through the notion of uncertainty.

Uncertainty

There are a few different concepts associated with uncertainty and a few of them have investing applications. I'm going to focus on one which is how it's often used in physics which is measurement uncertainty. 

Plantinga on Antirealism

So I decided to read Alvin Plantinga's How to be an Anti-Realist which I thought might be interesting. Unfortunately I was disappointed.

I won't present a thorough discussion of the paper but I will draw upon one bit of reasoning I found a bit goofy.

Is the Powerball lottery a good value?

The Powerball lottery jackpot for Saturday May 13, 2013 is estimated to be around \$600 million. That raises the question, with it being this high, is it a good gamble?

If math bores you, I'll give you a hint (it's not.) To show this is the case let's take a look at the payouts and the odds as listed here.

On the Use and Abuse of Debt

Debt plays an important role in the US economy (and the world economy for that matter). It's actually a very interesting social arrangement. I highly recommend David Graeber's book - Debt: The First 5,000 Years - if you're interested in exploring debt from an historical/anthropological perspective.(I may discuss this in a later post if I get around to it.)

Today I won't be questioning the existence of debt relations but will take them as given. The question here will be this: is there a sensible way to deal with debt?

The Perfect Parking Spot

So the other day I observed something I have many times before and it has always fascinated me (although it doesn't exactly surprise me.) As I'm walking toward the store to purchase groceries there's a gentleman in his vehicle waiting as a woman loads groceries into her vehicle. Now he's either checking out in a very stalker-like manner or he was simply waiting to take her parking spot. I would imagine he probably waited a least a minute or two to get this wonderful parking spot.

Now I could understand this under certain situations. For example, if available parking was limited then it might make sense to wait for a parking spot to be freed up. Or if the gentleman had some health problems that made walking long distances difficult. But I don't think either was the case here.

(1) There were parking spots, maybe, 20 spots away. So the gentleman was going to sit there and waste gas for a minute or two to alleviate himself from having to walk an additional 30 seconds more.

(2) If there were health issues, then I highly recommend he go see a doctor as it's not too difficult to get a temporary handicap parking pass. This would make available all of the, much closer, handicap parking spots.

But it really fascinates me that people (Americans) are willing to waste more time (and gas) looking for the perfect parking spot than just park somewhere in the back and walk a bit (and heaven forbid, burn a couple of calories.)

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On the Existence of Married Bachelors

In my series on That's an Empirical Question, I noted that philosophers often consider questions which are "empirical" as being outside the scope of philosophy. The domain of philosophy would then be some subset of "non-empirical" questions. Part of my goal has been an attempt to demarcate the empirical from the non-empirical.

Today I will continue this endeavor by exploring it from the other side: what makes a question non-empirical and is there a place for these sorts of questions. This is a very large subject, which could not be treated in one blog alone. I will, however, start with an example (as the title of the blog suggests): All bachelors are unmarried men.

Adam Smith on Equilibrium

The neoclassical conception of "equilibrium" is actually somewhat different than Adam Smith's characterization. In spite of that, neoclassical reasoning often (implicitly) assumes that their equilibrium  has the same implications as Adam Smith's. I would like to briefly describe Adam Smith's notion of equilibrium and would suggest that his notion is far more useful than the neoclassical market clearing equilibrium.

In  An Inquiry into the Nature and Cause of the Wealth of Nations (or simply Wealth of Nations),  Smith described two kinds of prices for commodities. This is all laid out in Chapter VII of Book I. Smith described the Natural Price and the Market Price of commodities. I will begin by looking at the latter as it's the closest concept to the current neoclassical understanding of equilibrium.

Competitive Advantage Types

Economic models often posit the existence of "perfect competition". But in the real world, companies are often able to establish a solid competitive advantage over other companies. This enables them to either be able to charge a higher price than competitors without significantly harming or sales, or it allows them to have lower costs thereby earning them higher profits by charging the same price (or a lower price to compete out competition).

In markets that exhibit "perfect competition", these advantages should be short lived. But in reality, many persist over long periods of time.

The key metrics to look at here are return on capital and cost of capital.  In perfectly competitive markets the two are equal. But sometimes a company can earn a higher return on capital than their cost. If this persists for a long time, it's an indication of a competitive advantage.

In Valuation: Measuring and Managing the Value of Companies, the authors (Koller, Goedhart and Wessels) list a large number of types of competitive advantages. I will reproduce the list here.

On the Subject Matter of Philosophy

Most disciplines have a very specific, known (and in some cases quite narrow) subject matter. The subject matter of the discipline specifies a domain over which research might take place. For biology, for example, the Greek origins of the term suggest the domain of study is life. Psychology is the study of the mind.

While it's not always the case we can define clear-cut boundaries, there are definite subject matters which we can point and say "that's a question that the psychologists should investigate".

This raises the question of what counts as the genuine subject matter of philosophy. In a broad sense, the term philosophy means "love of wisdom". It seems to encompass all that might be learned or discovered. For example, in its early stages, physics was often referred to as "natural philosophy".

While I am very much sympathetic to this idea, in practice, philosophers have tried to separate their discipline from other disciplines. So what is it that makes philosophy a unique discipline?

The Dollar is an Excellent Store of Value

To start off, I don't believe the title of this blog, at least not in the usual sense of how the dollar is defined. I would like to suggest, however, that the dollar has been, historically speaking, a good store of value. Whether or not it will do be so in the future is anyone's guess.

If you've ever had the "privilege"  of reading an economics textbook, it will give you a functional definition of money - money is defined in terms its functions. For example, according to the wikipedia article on money, money serves the following four functions:

1. Medium of Exchange
2. Unit of Account
3. Store of Value
4. Standard of Deferred Payment

The article notes that many economics texts do not list "standard of deferred payment", instead, treating it as parts of the other functions.

As far as the dollar is concerned, it's frequently acknowledged to satisfy 1, 2 an 4 but not 3.  I can use dollars to make all sorts of purchases (medium of exchange), all of the goods and services and debts are denominated in dollars (unit of account) and all of my debts can be dispelled by the use of dollars as a result of legal tender laws (standard of deferred payment).

But regarding store of value, the dollar has not served so well. As I illustrated in Figure 2 from my blog on Currency and Price Stability (reproduced here), the dollar has lost a significant amount of purchasing power over the years.

Einhorn and AAPL's preferred shares

So there's a lot of fuss regarding David Einhorn's proposal that Apple (AAPL) issue preferred shares in order to "unlock value".  To be clear, I like Einhorn and even liked his book, Fooling Some of the People All of the Time (in spite of the fact that it was excruciatingly detailed oriented).

But I concur with Prof Damodaran that this doesn't really create value; it simply changes capital structure. Granted, as Damodaran points out this might unlock price (I suspect it would since the market sometimes overprices leveraged situations. The fact that many analyses of Einhorn's strategy assumes that PE ratios will stay the same in spite of the leverage is evidence of this fact. They make the mistake that one wouldn't make if one understood the ideas in this post.)

But it does enhance value in at least one sense. Cash distributed now is worth more than cash sitting in an account earning next to nothing in interest. 

But I'm more interested in this question: is 4% the correct price for the preferred shares?

Understanding Enterprise Value

There's a procedure used when evaluating stocks of subtracting cash or subtracting "net cash". I don't think it's well understood so I'll give a brief presentation here.

The procedure is actually based upon two observations:

Grassmann's Real versus Formal distinction

So I'm currently reading A History of Vector Analysis written by Michael J. Crowe (or to say it another way, I'm not normal). I came across the book after reading this article: Hermann Grassmann and the Creation of Linear Algebra which I read because I have an interest in the development of linear algebra (aka I'm not normal).

Crowe quotes at some length from a book written by Grassmann entitled: Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durchAnwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik,Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert (which in German means tl;dr). But he makes the following distinction:

The primary division in all the sciences is into the real and the formal. The former represent in thought the existent as existing independently of thought, and their truth consists in their correspondence with the existent. The formal sciences on the other hand have as their object what has been produced by thought alone, and their truth consists in the correspondence between the thought processes themselves.

From what I'm gathering from what little bits Crowe has quoted, Grassmann thinks that his system (which from what I gather was an early form of modern day linear algebra) as being merely "formal". It's existence has been posited through pure thought but it doesn't correspond to anything in the world (outside of the correspondence to human thought). Contrast that with geometry which corresponds with space and is therefore an example of a "real" science. He even went so far as to suggest that geometry properly ought not be included within mathematics:

It had for a long time been evident to me that geometry can in no way be viewed, like arithmetic or combination theory, as a branch of mathematics; instead geometry relates to something already given in nature, namely, space.

What he calls "pure mathematics", by contrast is:

... the science of the particular existent which has come to be through thought. The particular existent, viewed in this sense, we name a thought-form or simply a form. Thus pure mathematics is the theory of forms.

I'm not sure how far these distinctions would go today. Consider space for example. In a post-Einstein world, space has different structure when in the presence of objects of mass. Mass actually bends or warps space. And it's not entirely restricted to 3 dimensions as it was in Grassman's day.

On the more "abstract" side of things, the lines are even more blurred. We now deal with infinitely dimensional complex vector spaces to explain such phenomenon as matter waves.

At this point there should be a cue for a joke about Hilbert spaces but I'm too lazy to write one. So I'll just get a room in his hotel and crash for the night.

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What is a Financial Bubble? Part I

The idea that drives this metaphor is blowing a bubble. One can blow it larger and larger but eventually it pops. Some economists have denied the existence of bubbles while others have expressed skepticism that the concept is well-defined enough to be scientifically meaningful. Eugene Fama, a prominent supporter of the Efficient Markets Hypothesis, stated as much in an interview in The New Yorker.

I share these reservations over defining a bubble but I still suspect it's a useful concept. As a result, I'd like to explore the issue a bit. I'll be doing so in a series looking at various examples and aspects of financial bubbles in order to get a better grip on whether or not we can give it a meaningful definition.

Gettier Intuition

One common criterion used to determine whether or not something is knowledge is the "justified true belief" criterion (JTB). It has many forms but the basic idea is this:

S knows that P means:

1. S believes that P.
2. S has justification for believing P.
3. S is true.

In 1963 Edmund Gettier proposed counterexamples to the above relationship. The idea is pretty simple. Construct a scenario in which (1), (2) and (3) are satisfied, but it fails to be knowledge. We'll look at an example from his original paper: Is Justified True Belief Knowledge? In particular, we'll look at Case I.

Downside Risk Investing

I'd like to give some consideration to a class of investment strategies which I call Downside Risk Investing. These strategies earn returns by taking on downside risk while at the same time having limited upside potential. There are a number of strategies that fit this bill which I'll outline below giving several examples.

There's a metaphor that goes around talking about "picking up pennies in front of a steamroller". It's in reference to investment strategies in which the investor risks getting run over by a steamroller (taking the risk of getting wiped out) for the benefit of receiving a couple of pennies. This entire reference is to Downside Risk Investing.

Now in some sense, many investment strategies have the potential for huge losses with limited upside. The question we should be asking is this: how many pennies do I need to be able to pick up for a particular Downside Risk Investing to be a good strategy? I don't know if I have an answer to that question but I'd like to present a discussion. So without further ado, here are some examples of strategies that I think fit the bill (some of which may be good strategies to employ).