Financial Mathematics Text

Wednesday, December 17, 2014

Some Ambiguities Regarding the Cost of Capital

So I'd like to suggest there's an ambiguity in measuring the cost of capital. As far as I can tell, there isn't always a straight forward consensus on how this should be handled. There are a number of arguments in favor of particular positions.

Today I'm going to very briefly present the issue.

Friday, November 21, 2014

ROE, ROA and Leverage (Update)

Sometimes I don't see alternative ways of writing expressions until after the fact. So this will just be a brief modification of a previous post: Relating ROE with ROA and Leverage.

Monday, November 17, 2014

Option Returns - Empirical Results

Previously, I looked at what we'd expect call and put options would be if we assumed that stock returns follow a normal distribution (see the Expected Return of a Call Option and Put Option).

My findings indicated that the underlying assumptions of the Black-Scholes pricing model are inconsistent with the mean-variance view of risk. This was not an empirical result, mind you. Empirically, I've yet to find a single set of financial data that was normally distributed. It was a theoretical result; the theory is inconsistent with a mean-variance view of risk and return.

Today I'll be looking at some odd empirical results. I wanted to see what actual returns actually looked like. As it turns out, they're even worse than what the theory predicts.

Friday, November 14, 2014

On Counting (Exploring Operational Definitions Part III)

For the first two parts in this series see:

Exploring Operational Definitions: Part I
Exploring Operational Definitions: Part II - Distance

Perhaps the "simplest" procedure that most folks have learned is the technique(s) of counting. What I would like to explore is that there are a variety of techniques that we call counting. In some cases they build on one another. In other cases, they are techniques which give "approximate" solutions.

Of course not all societies count things (see here). Nonetheless, I suspect that many of our "intuitions" about mathematics ultimately stem from our earlier experience with counting. Our attachment to such intuitions will somewhat determine how willing we are able to accept alternative definitions and techniques for counting. Today I'll explore a few of these definitions.

Monday, November 10, 2014

Financial Mathematics: Statistics - Moments

In statistics, there are a variety of calculations referred to as moments. We'll be discussing three types of moments: Raw Moments, Central Moments and Standardized Moments.

Monday, November 3, 2014

Some Comments on the Quantity Theory of Money

As I mentioned in my previous post - Inflation - Why the official numbers are wrong! - I pointed out that the general theory for which inflation is based upon implies that the "price level" is a vector but measures of inflation represent this as a scalar. Today I want to explore how that complicates the picture for the Quantity Theory of Money.

Friday, October 31, 2014

Financial Mathematics: Statistics - Expected Values

In statistics, a probability distribution is any function, $f(x)$ which is never negative (probability is either 0 or positive) and it sums up to 1 (100%). Mathematically we'd express this as:
\forall x f(x)\ge 0 \\
\sum_x f(x) = 1
In the case of a continuous random variable, the second formula would be expressed as an integral:
$$\int_x f(x)dx = 1$$
There are some differences between discrete and continuous random variables but the ideas behind them are the same.

To motivate the idea behind an expected value, we'll begin with a more familiar concept: an average.

Tuesday, October 28, 2014

The Efficient Markets Hypothesis is Meaningless

I'm going to begin a critique of the Efficient Markets Hypothesis (EMH). This is not the first nor will it be the last that have been presented. Most of these critiques accept the basic paradigm an attempt to empirically prove that one can "beat the market".

For the practitioner, this can be quite appealing as it allows one to find some strategy that would allow one to earn "excess returns".

My approach, which I've been toying with in my mind for the last year or so, is going to be a bit different. My contention is that the entire paradigm is questionable and perhaps "meaningless"1. At the very least, proponents of EMH have a a lot more work to do as there is a lot of ideological baggage and not much in the way of a legitimate scientific hypothesis.

Monday, October 20, 2014

How to Ignore the Noise in Financial News

One of the most difficult things we face in the information age is the problem of too much information. It's everywhere around us. There's absolutely no way for us to get through all of that information much less be able to utilize it.

There's even a good deal of research that indicates that, not only are we unable to handle extra information, that additional information may make us less accurate and more confident in our inaccurate predictions: The illusion of knowledge: When more information reduces accuracy and increases confidence.

Now it seems to me that there are at least three goals we need to focus on in order to handle all of this information:
  1. Focus on important information.
  2. Ignore the useless noise.
  3. Know what we do not know.
While I think this is important in general terms, there is the question of how to deal with financial news. 

Monday, September 22, 2014

Cape Alternative - Update

In a previous post, CAPE - An Alternative Calculation, I discussed some problems with CAPE (cyclically adjusted price to earnings ratio) and offered a solution which addresses one of those problems which is the growth problem. Today I'll briefly illustrate the growth problem and then look at my solution from an historical perspective.

Tuesday, September 2, 2014

More on the Constant Growth Assumption

So earlier, in Uncertainty of the Constant Growth Assumption, I discussed how even if we know with certainty (hah!) what rate of growth cash flows (in our example, dividends) will grow at, we may still end up getting the value incorrect. This is due to the fact that actual cash flows are lumpy and don't grow at a nice constant rate but vary quite a bit. The result is that the present value of the asset can be a good deal more or less than where we estimated it to be.

Today I want to look at this from a different perspective. I also wanted to change some of the inputs of the model I used (which, you could do on your own if you downloaded or made a copy of the original spreadsheet.) Instead of focusing on how the actual present value of the cash flows differs from what the model predicts, I want to look at it from the perspective of what returns you'll actually get.

Tuesday, August 12, 2014

O Captain! My Captain!

So August 11 was a sad day apparently; I managed to misplace my keys.

In other news, I found some of the movies you can find on Netflix and Amazon Prime starring a Mr. Robin Williams.

Monday, August 11, 2014

Inflation - Why the official numbers are wrong!

I'm certainly not the first to claim this and I won't be the last. But I think I'll say this differently than most who happen to say it.

For starters, when I refer to inflation, I'm preferring to price inflation.  Some (particularly Austrians), by inflation, mean monetary inflation so I think it's important to be clear on what's being discussed.

Many feel that the current inflation figures (whether we're talking CPI or CPI minus food and energy) understate the actual rate of inflation. Other methods are sometimes used which in many cases are not any better than the official numbers (and in many ways worse.)

Howerever, what I'd like to suggest is that Any method of coming up with an inflation number is flawed. To understand this we'll have to take a brief detour.

Friday, July 11, 2014

Is AMD a better value than Intel?

Bruce Greenwald, in his book Value Investing: From Graham to Buffett and Beyond, devoted an entire chapter to Intel and its competitive advantages. Of particular note was its economies of scale which permitted it to spend less on research & development than AMD as a percentage of sales.

That appears to still be true today but less so:

Friday, July 4, 2014

How to Use More than 10 Percent of Your Brain

Every now and then I happen to catch some regular good 'ol fashion TV. Such events are not as common now as I typically stream content from the likes of Netflix and Amazon Prime if I'm going to sit myself down to be entertained.

Anyway, I happened to catch a commercial for an upcoming movie called Lucy. Here's the trailer:

Monday, June 30, 2014

The Expected Return of a Put Option

In my previous post, I explored The Expected Return of a Call Option by assuming that stock price returns follow a normal distribution. I then looked at some attributes of the distribution of option returns under that assumption.

Today I'll apply the same technique to put options.

Thursday, June 26, 2014

Some Recent Updates to AF2P

So I've made some changes recently. Here's a quick overview of some of the changes (feedback welcome).


So I've changed the color scheme a bit. It looks different (better?) Take it for what it's worth (probably not much; I'm pretty lazy when it comes to aesthetics.)

Share Buttons

I've changed the share buttons a bit. If any of them don't work let me know.


These are located beneath the share buttons. They allows you to check a "reaction" to a blog without responding. If you have ideas on more reactions you'd like to see, let me know and I'll consider adding them. (I wanted to add something like "garbage" but I wanted to use a better term than "garbage". Feel free to offer suggestions.


So I was recently informed that my comments section was restricted to those only with a blogger account. While I still want to avoid anonymous posters, I did expand the reach a bit. You can now respond with a blogger, WordPress, AIM (1990's anyone?), LiveJournal, TypePad or OpenID account. The latter should open up many options including Google and Yahoo accounts (see here.)

Subscription Options

There are a variety of subscription options. You can still use feedburner which allows you to subscribe via any RSS feeder or email. In addition, my twitter and facebook page auto-post items from feedburner so following either of those would suffice as well.

Monday, June 23, 2014

The Expected Return of a Call Option

If I were to buy a call option, what return should I expect to get? Is it worth buying? Or is it a losing proposition?

Today I hope to address this question from one angle. What does standard finance theory actually have to say about the matter?

Now of course standard finance theory may very well be wrong. Some of the assumptions that will be used today are problematic.

As it turns out, this presentation will be a sort of a proof by contradiction. I believe it will show that standard finance theory is inconsistent. Or at a minimum, certain propositions that are often assumed in various financial models are incompatible depending upon interpretation.

While my interest in this is somewhat academic (aren't all of my interests?) there may be a practical application to this which has to do with estimating the cost of equity. It may be possible to use this device as an alternative method of determining cost of equity (say, versus the lousy CAPM model.)

But that will have to be worked out some other time. Today I just want to show how one can calculate the expected return on a call option.

Sunday, June 15, 2014

Real Estate: The Better Inflation Hedge

In my previous post, I discussed having a healthy dose of skepticism when one sees a correlation between two variables. The relationship may not hold very well out of sample (such as future performance).

But I also discussed gold as an inflation hedge and I suggested that gold was a lousy inflation hedge in spite of what most people seem to believe.

Today I want to discuss another asset class which, although may or may not be a good inflation hedge, is nonetheless a better inflation hedge than gold. Furthermore, this asset class has many other attractive features that suggest it should be pursued before gold if you're looking for a hedge against inflation.

Monday, June 9, 2014

Gold, Hedges and Correlations

Today I'm going to touch on a sort of theme I have here and that's regarding on how to assess correlated data. How much can we actually read into it? How do we determine that there's a fundamental relationship that, not only holds well in the past but will continue to do so in the future?

To do that I'll be taking a quick look at gold and why I consider the "inflation hedge myth". In short, gold is not an inflation hedge. At least not on any reasonable time scale.

Thursday, June 5, 2014

Relationship between VIX and Credit Spreads

This is motivated by a post from David Merkel of the Aleph Blog, called Buy Stocks When Credit Spreads are High, Sell When They are Low.

Merkel suggests that "credit spreads and implied volatility are cousins. When there is complacency, both are low. When there is panic, both are high."

Monday, June 2, 2014

Exploring Operational Definitions: Part II - Distance

So in Part I, I began discussing operational definitions. In particular, I've been focusing on the concept of distance.

I proposed that we might consider our intuitions regarding answering the question - what is distance? - by means of early experience of operational definitions of distance. The idea being that what we consider distance is going to be closely related to ways we were taught to measure distance.

Friday, May 30, 2014

Misnomers in Finance

You know what really grinds my gears1. . .

                                                                 . . . misnomers in finance.

Essentially there is some technical jargon that I think was inappropriately named. Here are a couple of examples.

Monday, May 19, 2014

Ode to Lady Luck

In Roman mythology, Fortuna was the goddess of luck and fortune.

Monday, March 24, 2014

Uncertainty of the Constant Growth Assumption

In Uncertainty and Margin of Safety, I discussed the role uncertainty plays in valuation. Today I want to take a look at one of the assumptions built into many discounted cash flow (DCF) models. That assumption is the constant growth rate assumption.

Monday, February 24, 2014

Minimum Wage and Inequality

One topic that is an interest of mine is the topic of growing income inequality. Over the last 30 years or so, a small fraction of the population in the US has increased its share of income (and even more wealth) while the middle class has been gradually gutted and those on the bottom have been stepped on. Ultimately I don't think it's a sustainable situation and something needs to be done to address those issues.

Today I want to look at a hot topic in politics: the minimum wage. President Barrack Obama has been pushing an increase of the Federal minimum wage to $10.10. And there's plenty of resistance to that but I don't want to focus on that too much. We can really consider two issues:

(1) Is raising the Federal minimum wage a good way to alleviate income inequality?
(2) Assuming (1), where should the Federal minimum wage be?

It's the latter question I'd like to take a look at today.

Thursday, February 13, 2014

Return on Invested Time: Developing Understanding

Time is our most valuable possession; and, yet, we trade it for other things.
One thing I noticed relatively early on is my ability to process information better than others. I don't know to what extent this is genetic or learned but I do think part of it comes down to how I engage information.

Many tasks in my early education experience I considered to be pointless or a waste of time. While I jokingly referred to myself as a slacker, to some extent it was deliberate in that I saw some activities as being unproductive and would rather spend my time doing activities either more enjoyable or more productive.

In hindsight, I think I may have misjudged in some cases but in many cases my approach has been successful. Here's a look at my overall approach.

Thursday, January 30, 2014

McDonald's and Wages

There's a considerable amount of debate revolving around the minimum wage and the vast number of workers in the US that earn a very low wage. I'm personally of the view that the growing inequality in the US in which the top 1% (and 0.1% which looks even worse) have been taking a larger share of the income.

A good portion of the talk has concerned fast food workers of which McDonald's is the largest and most focused on.  Today I want to take a practical look at McDonald's and wages.

Monday, January 20, 2014

CAPE - An Alternative Calculation

Today I want to look at an alternative way to calculating the Cyclically Adjusted Price to Earnings Ratio or CAPE for short. The standard approach to CAPE suffers from a few drawbacks and I think the calculation I'm proposing can address some of those drawbacks.

Thursday, January 9, 2014

My Failed Inflation Prediction

In August 2009 I made some tongue-in-cheek type predictions. To be clear, I roughly believed them but at the same time I knew they were some gut feelings based on some rough models I had in my head.

My inflation prediction was that that CPI would hit 250 by Fall 2011. It was about 215 at the time which put inflation to be close 8% a year. Apparently we're still not there.

Monday, January 6, 2014

Correlation as a Substitute for Critical Thinking in Finance?

I frequently come across graphs like these on various blogs and articles. I'm going to single out one, not because there is anything particularly wrong about it (they're all wrong), but only to use it for illustrative purposes. I could really pick out any of these and offer the same criticisms.

This one I found in the article The Declining Inflation Expectations Chart That Should Have Stock Investors Very Concerned. The original chart apparently come from Dan Greenhaus via twitter.

Friday, January 3, 2014

Interesting Approach to Predicting Future Stock Returns

So Jesse Livermore (the name on his twitter account, which comes from the trader who is known for his Reminiscences of a Stock Market Operator) has an interesting blog called Philosophical Economics. Under discussion today is an interesting approach to predicting future stock returns. The blog post under question is entitled The Single Greatest Predictor of Future Stock Returns.

In what follows I'll offer a brief summary of the post (he writes longer blog posts than I do!) and a few points of criticism as well. The criticisms offered, I think, will be in the spirit of Jesse Livermore's criticisms of other metrics which attempt to predict future returns (see here).

Wednesday, January 1, 2014

2013: Year in Review

So I started this blog several years ago and didn't do much with it. I picked it up again in 2012. 2013 was the first year I blogged the entire year.

Starting out this was primarily a philosophy blog. I still think of it as such but I've devoted several thousand words to finance as well. My guess is that most of my readers (you know, the three of you . . . well one since two of you are just voices in my head) are more interested in finance than philosophy but I think there's a lot of value that can be gained from philosophy.1 You'll find a good number of my finance posts are still related to methodological questions.