Financial Mathematics Text

Saturday, October 26, 2013

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.

Thursday, October 24, 2013

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.

Saturday, October 19, 2013

Some Thoughts on Risk/Return Tradeoff and EMH

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

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

Sunday, October 13, 2013

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.

Wednesday, October 9, 2013

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. 

Tuesday, October 8, 2013

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).

Saturday, October 5, 2013

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:

Thursday, October 3, 2013

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:

Tuesday, October 1, 2013

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.