Financial Mathematics Text

Monday, December 31, 2012

New Year's Resolutions - 2013

It's a tradition around the New Year to set "New Year's Resolutions". The idea is to reflect on your life and the past year and set goals to achieve certain things or improve oneself. I suppose in principle it's not a bad idea but in practice most people actually fail.

For starters, many people make rather vague goals such as "I'm going to exercise" or "I'm going to be a nicer person". It's not entirely clear what would count as "accomplishing" such goals.

A better approach is to make specific, achievable goals such as "I want to lose 10 lbs by the end of the year" or "I'm going to donate $1000 to charities". These are measurable goals so that when you fail in a couple weeks (and you know you will!) there will be no denying it.

So enough of this nonsense introduction. Here are my resolutions for next year.

Thursday, December 27, 2012

Who are the 1%?

In Income Inequality in the US, I noted that the top 1% (and especially the top 0.1%) have been taking a greater share of productivity gains over the last 50 years (especially since the 1980's).

But that raises the question who are these 1%-ers and why have they been able to take a larger share of productivity gains?

We'll start by looking at some data found in this study which gives a breakdown of the top 1% and top 0.1% for years between 1979 and 2005. Here's a summary of the 2005 data:

Sunday, December 23, 2012

Deriving Graham's Intrinsic Value Formula

In Chapter 11 of Benjamin Graham's The Intelligent Investor, Graham introduces a formula for valuing growth stocks.

$$P = E (8.5 + 2G)$$

where P is the value or price, E are current (or normal) earnings and G is the growth rate. 

In a footnote, Graham remarks:
Note that we do not suggest that this formula gives the "true value" of a growth stock, but only that it approximates the results of the more elaborate calculations in vogue.
He mainly used the formula to show how absurd some implied growth expectations are. In other words:

Thursday, December 20, 2012

That's an Empirical Question: Part IV

This is the 4th part in a series called "That's an Empirical Question". For the other parts of this series see here:

Part I
Part II
Part III

I think this one deserves an appropriate subtitle.

Is Induction Always Empirical?

In Part II, I suggested that induction is an important "empirical tool". Assuming that's true, that still raises the question whether all uses of the "induction tool" necessarily amounts to an "empirical investigation".

To explore this, as per usual, I will use an example: The Goldbach Conjecture.

Wednesday, December 19, 2012

Dave Ramsey and Asset/Liability Matching

While I don't personally follow Dave Ramsey's advice, I do occasionally read his column in the Sunday paper. In spite of that, generally I think it's good advice and more people would do well to follow it. With private debt levels at very high levels right now, especially household debt, it makes sense to take a close look at how we use debt and whether or not its appropriate.

His most recent post gave me pause for concern though. A family has some money saved in a mutual fund for their son's college tuition. Dave recommends keeping the money there "until right before you write the checks." His rationale is that he returned 16% last year and he's hoping to get at least another 10% this year.

This is not sound advice. The reason comes down to asset/liability matching.

Tuesday, December 18, 2012

Is Housing an Investment?

The conventional wisdom says that it is. I'd like to suggest it's a little more complicated than that. Rather than suggesting that housing is an investment, I will argue that housing can be an investment and spell out under what conditions it is.

To some extent I will be relying on the Graham & Dodd definition of an investment. According to Graham & Dodd (from Security Analysis):
An investment operation is one which, upon thorough analysis, promises safety of principal and a satisfactory return. Operations not meeting these requirements are speculative.
 Regarding speculation, two types of speculation are classified (Graham & Dodd):
  1. Intelligent Speculation - The taking of a risk that appears justified after careful weighing of the pros and cons.
  2. Unintelligent Speculation - Risk taking without adequate study of the situation.
Most home buyers would classify as not giving "adequate study of the situation".

So when is a house an investment? 

Monday, December 17, 2012

That's an Empirical Question: Part III

This is a follow-up in a series exploring what the difference between "empirical" versus "mathematical" versus "philosophical" questions. For the other posts in the series see:

That's an Empirical Question: Part I
That's an Empirical Question: Part II

I'd like to consider a thought experiment though I think you can probably find some similar at manufacturing firms in the real world.

Sunday, December 16, 2012

Income Inequality in the US

Here's income inequality in one simple chart.

Friday, December 7, 2012

Are stocks a good value relative to alternatives?

To explore the question about stocks we'll need some criteria and we'll need to look at some various metrics. For simplicity, I will be assuming that "stocks" means the "market" and more specifically the S&P 500. All data and conclusions will be relative to that. The main alternative investment we'll look at are Baa rated corporate bonds.

Sunday, December 2, 2012

Do philosophers know particulars?

In Methodists Vs Particularists, I suggested that philosophers who are particularists may not, in fact, be so. It's possible that there are no particulars that are known per se but rather a broad set of "methods" that are employed. This may not be explicit criterion per se; as a result I shall refer to these simply as "heuristics".

We'll consider a common "simple" example in philosophy: color identification. I believe that most children learn color identification via ostensive definitions. Consider the following scenario:

Thursday, November 22, 2012

Methodists Vs. Particularists

In Intuition in Philosophy, I suggested that one difference between, say, mathematicians and philosophers is that philosophers will willingly privilege an intuition over a derived result whereas the mathematician will reject the intuition in favor of the derived result. To some extent I think this characterization may not have been entirely accurate. At the very least some clarification needs to be made.

I here will argue that many philosophers are particularists whereas mathematicians tend to be methodists. "Intuition" plays a role in what particulars the philosopher claims to have knowledge. This allows for there to be something along the line of "mathematical intuition" of which I think is believed to be present by many mathematicians. I will briefly sketch out the issue here.

Tuesday, November 20, 2012

On the Existence of Santa Claus

While this is a very broad topic which this one blog cannot do justice, I would like to consider one argument against the existence of Santa Claus in more detail. The strategy will be to show that this argument is inadequate given two considerations.

The crucial piece of evidence for this argument is a statistical tendency for children of wealthy families to receive gifts totaling a higher value than children of poor families. This leads to two general claims.

Wednesday, October 17, 2012

Was Chess Invented or Discovered?

One common comparison made within philosophy of math circles is a comparison between the game of chess and mathematics. Variants of, in particular, formalism hold that mathematics is a sort of game with well defined rules that mathematicians play. I think the game of chess is actually quite apt for the comparison. Like all metaphors, it has limitations but I think it's still an interesting one.

The closest comparison, in my view, is between chess and geometry. Here's how it works.

Monday, October 15, 2012

Yield-Duration Curve

Most people are familiar with the "yield curve" which compares yield with maturity. I personally think a more useful curve looks at yield and duration.

Duration is a metric which gives a rough indicator of how much a change in interest rates will affect the price of a bond. Roughly speaking, if a bond has a duration of 10 years, a 1% change in interest rates (say from 4 to 5%) will result in a roughly 10% change in the price of the bond. So duration gives one information about interest rate risk.

Sunday, October 14, 2012

Friday, October 12, 2012

Being Without Numbers

Linguistic anthropologist Daniel Everett has been researching and living with (off and on) an Amazonian tribe called the Pirahã for a few decades now. In Don't Sleep, There Are Snakes: Life and Language in the Amazonian Jungle, Everett gives a semi-autobiographical account of his interactions with the Pirahã.

Wednesday, October 10, 2012

Intuition in Philosophy

Intuition has an interesting role in philosophy, particularly analytic philosophy. One of its functions is epistemic. It's used to test various assumptions and conclusions in philosophy for validity or truth. If something fails to stand up to intuition the propositions in question get dismissed.

There's also a good deal of literature in a field known as "experimental philosophy". The research often takes the form of surveys to different groups of individuals (intro to philosophy students, professional philosophers, etc). The purpose of the surveys is to assess what "intuitions" people have. What the research tends to indicate is that intuitions vary greatly between groups depending upon a variety of circumstances (culture, philosophical exposure, etc.)

Monday, October 8, 2012

Currency and Price Stability

There are two related claims made by gold standard advocates. These claims are:


1) Under a gold standard, prices are stable.
2) Prices ought to remain stable.

I) We'll start by looking at the first claim. What does it mean to say that prices are stable?

Saturday, September 8, 2012

Modeling Gold Returns: A Case Study

In my previous post, I considered the problem of induction. Looking at emeralds and noting that they are green (or grue) is supposed to be a simple example since making color judgements is presumably a simple task. The example could be further complicated if we factor in the vagueness of making color judgements. After all, it's likely the case that the distribution of electromagnetic frequencies differ between one emerald and the next. (And they would differ further depending on the distribution of EM frequencies of the "white light" used upon it.)

In most real world scientific inquiries, these uncertainties are present and need to be dealt with. So instead of focusing on the sorts of examples used in "simple" philosophical thought experiments, I thought I would provide a more detailed example.

The motivation for looking at this came from an interesting paper entitled The Golden Dilemma. I will make frequent reference to Exhibits from this paper. [1]

Monday, September 3, 2012

The Problem of Induction

This is a continuation of a series devoted to the question of what an "empirical question" is. See Part I and Part II if you're interested.

One tool that is considered essential to the empirical sciences is induction. It's importance is, in my opinion, overstated by most philosophers (I firmly believe that abduction rarely gets the proper credit it deserves). In spite of that, I do not dismiss its important role in scientific inquiry.

That's an Empirical Question: Part II

See here for Part I.

Thus far I've conjectured that an empirical question in some way or form entails the use of empirical tools as distinct from other kinds of tools. One type of alternative tool is mathematical while another is philosophical. This interpretation rests on the idea that (some) philosophers believe there to be a distinct set of questions which can be addressed only with a particular set of tools. I would like to further explore this issue.

Monday, July 30, 2012

That's an Empirical Question: Part I

Having studied philosophy and having discourse with many philosophers, I had heard (and used) this phrase on many occasions. At times during the course of a conversation a conjecture or question is generated and it's decided that the conjecture or question can (should) be decided empirically. After demarcating the question into the "empirical" pile, philosophers then move on to other questions, discussions, etc.

My goal today is to offer a brief overview of two questions to which I will offer hypotheses regarding them (whether or not my questions and/hypotheses are "empirical" can be decided by the reader.)