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

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.

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.

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.

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).

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.

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

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.

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.

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:

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.

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.

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

loading..