Financial Mathematics: Statistics - Properties of Distributions

Probability distributions have a number of properties which help us summarize and characterize them. We'll look at some of these properties, how they are calculated and what they are used for.

Contents:

I) Mean
II) Median
III) Variance
IV) Skewness
V) Kurtosis

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.

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.

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.

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.

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:
\[\begin{align*}
\forall x f(x)\ge 0 \\
\sum_x f(x) = 1
\end{align*}\]
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.

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.

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. 

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.

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.

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:

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.

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.

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.

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.

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."

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.

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.

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.

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.