- Probability Review
- Linear Algebra Review
- The Normal and Lognormal Distributions
- Random Walks
- An Introduction to Utility Theory
- An Introduction to Portfolio Theory
- Portfolio Optimization: Part 1 - Unconstrained Portfolios
- Portfolio Optimization: Part 2 - Constrained Portfolios

- Savings Rates and "Economic Security" in Retirement
- Financial Planning Using Random Walks
- Asset Allocation and Portfolio Survival
- More Portfolio Survival Studies

I like to study finance as a hobby. Every once in a while I write something about what I'm learning and add it to this web page. The reason I do this is because I learn best when I write up what I'm learning as if I were trying to explain it to someone else. These are really private notes that I wrote for my own benefit as a student, but because I have no shame, I am making them available to anyone else who might be interested here on my web site. Perhaps nobody will be interested, but that's OK. The discipline of writing up formal papers about these topics is an enormous aid in helping me learn the material, and that's the point of the exercise.

### Modern Portfolio Theory

### Financial Planning

### Option Pricing

### Miscellany

### Popular Articles

Because my profession is software development, I'm particularly interested in using computers to implement the theories and models of mathematical finance both to help visualize the mathematics and to apply the theories and models to practical problems.

There's lots of math in most of these papers. I assume that the reader is comfortable with basic undergraduate level calculus and probability theory. If you don't like math, you won't find my papers interesting or useful. On the other hand, if you're a serious advanced student or practitioner of mathematical finance, you've also most likely come to the wrong place. This is all very basic beginning theory, not original research or new results. You might, however, find the papers useful if you're a beginner like me trying to learn this material, if you're interested in ideas for teaching the material to beginners, or if you're interested in using computers to implement some of the theories and models for practical applications.

The Popular Articles are exceptions. They are attempts at popular presentations without lots of math and may be of more general interest.

All of the papers are in Adobe Acrobat PDF format except for the popular articles, which are plain web pages.

This series of homework papers is an attempt to derive the major results of Modern Portfolio Theory from first principles, with as much detail and mathematical rigour as possible. Although the presentation is formal and has lots of math, there are also many examples to illustrate the ideas being discussed, and I have tried to discuss the fundamental economic principles underlying all the equations (it's the point of the exercise, after all). The papers should be read in order. The later papers assume that the reader is familiar with the earlier ones.

This is in a sense the "textbook" I wish I had when I first started studying finance in 1998. The undergraduate and MBA textbooks on finance and investing are OK, but they avoid the math, and seem to be proud of it. The academic papers and books are OK, but they are way too advanced and assume way too much. I've never been able to find something in-between that met my needs and suited my (perhaps peculiar) tastes, so I had to invent it myself. (The popular books and articles on finance and investing are total trash and should be avoided like the plague, with a very few notable exceptions.)

September 11, 2002. Last updated November 3, 2011.

We define and review the basic notions of variance, standard deviation, covariance, and correlation coefficients for random variables. We give proofs of their basic properties.

September 11, 2002. Last updated November 3, 2011.

A review of the notions of matrix singularity and solutions to simultaneous systems of linear equations, with proofs.

The Normal and Lognormal Distributions

February 2, 1999. Last updated November 3, 2011.

The basic properties of the normal and lognormal distributions, with full proofs.

January 28, 2005. Last updated November 3, 2011.

We develop the formal mathematics of the lognormal random walk model. We start by discussing continuous compounding for risk-free investments. We introduce a random variable to model the uncertainty of a risky investment. We apply the Central Limit Theorem to argue that under three strong assumptions, the values of risky investments at any time horizon are lognormally distributed. We model the random walk using a stochastic differential equation. We define the notion of an ``Ito process'' and prove that it is equivalent to our formulation.

We apply the model to the S&P 500 stock market index as an example. We learn how to do parameter estimation for the model using historical time series data and how to do calculations in the model in computer programs. We discuss how uncertainty and risk increase with time horizon when investing in volatile assets like stocks, contrary to popular opinion.

We conclude by asking the all-important question of how well the simple random walk model describes how financial markets actually work. We mention known failings of the model and conclude that at best it is a rough approximation to reality and should be used for real-life financial planning with caution.

An Introduction to Utility Theory

March 29, 1999. Last updated November 3, 2011.

A gentle but reasonably rigorous introduction to utility theory.

Utility functions give us a way to measure investor's preferences for wealth and the amount of risk they are willing to undertake in the hope of attaining greater wealth. This makes it possible to develop a theory of portfolio optimization. Thus utility theory lies at the heart of modern portfolio theory.

We develop the basic concepts of the theory through a series of simple examples. We discuss non-satiation, risk aversion, the principle of expected utility maximization, fair bets, certainty equivalents, portfolio optimization, coefficients of risk aversion, iso-elasticity, relative risk aversion, and absolute risk aversion.

Our examples of possible investments are deliberately over-simplified for the sake of exposition. While they are much too simple to be directly relevant for real-life applications, they lay the foundation upon which the more complicated relevant theories are developed.

This paper was inspired by the first few pages of chapter 2 of Robert Merton's book "Continuous Time Finance." It is a student's clumsy attempt to fill in the gaps in Merton's presentation of this material. Merton assumes basic utility theory as a given. For your ignorant author it was anything but a given. Our notation and terminology are largely that of Merton's.

An Introduction to Portfolio Theory

April 10, 1999. Last updated November 3, 2011.

We introduce the basic concepts of portfolio theory, including the notions of efficiency, risk/return graphs, the efficient frontier, iso-utility (indifference) curves, and asset allocation optimization problems. We develop the theory in both a simplified setting where we assume that returns are normally distributed and in the more palatable random walk model where returns are lognormally distributed.

Portfolio Optimization: Part 1 - Unconstrained Portfolios

September 11, 2002. Last updated November 3, 2011.

We recapitulate the single-period results of Markowitz and Sharpe in the context of the lognormal random walk model, iso-elastic utility, and continuous portfolio rebalancing. We formally derive the solution to the unconstrained optimization problem and examine the mathematical properties of the resulting efficient frontier and efficient portfolios. We derive the two-fund separation theorem both in the presence of a risk-free asset and in its more general form. We derive and briefly discuss the Capital Asset Pricing Model (CAPM). We present several examples parameterized using market return data for US stocks, bonds, cash, and inflation.

The Microsoft Excel spreadsheet that was used to prepare the examples is also available.

We also have a simple Excel spreadsheet that can be used to calculate optimal portfolios and graph efficient frontiers in the two, three, and four asset cases.

Portfolio Optimization: Part 2 - Constrained Portfolios

March 28, 2005. Last updated November 3, 2011.

We develop the critical-line algorithm for solving portfolio optimization problems with low and high bound asset constraints.

The program PortOpt is a companion to this paper.

These papers are more applied than the theoretical ones in the MPT series. They focus on applying the theories to the practical problems of financial planning, and retirement planning in particular.

Savings Rates and "Economic Security" in Retirement

November 3, 1998. Last updated November 3, 2011.

Some simple number crunching using historical market return data for retirement planning. How much do we need to save to provide for a comfortable and secure retirement?

Financial Planning Using Random Walks

February 3, 1999. Last updated November 3, 2011.

We develop an enhanced random walk model for retirement planning. Our model includes a new term to accommodate periodic additions to or withdrawals from a portfolio. We deal with the problems of modeling social security, salary growth, inflation, investment expenses, and asset allocation among cash, bonds, and stocks. We investigate a basic problem of post-retirement planning, the risk of outliving your money. We use a Monte Carlo simulation technique to implement the model in a computer program.

The program Random Walker is a companion to this paper.

Asset Allocation and Portfolio Survival

February 15, 1999. Last updated November 3, 2011.

We examine the impact of asset allocation on the problem of outliving your money in retirement. Given a real (inflation-adjusted) withdrawal rate, we run a large number of Monte Carlo simulations to determine the asset allocation which maximizes the 30 year survival rate. We see how the composition of this optimal portfolio changes as the withdrawal rate changes.

More Portfolio Survival Studies

March 6, 1999. Last updated November 3, 2011.

We examine the impact of time, investment expenses, expected return and volatility on the portfolio survival problem.

The risk of outliving your money in retirement increases rapidly with time horizon and does not begin to level off until very long horizons.

Investment expenses are arguably the single most important factor affecting survival rates. Retirees who wish to minimize the risk of outliving their money are well-advised to use low-cost index funds and, if possible, forgo the services of expensive investment advisors.

Portfolio efficiency has a significant impact on survival rates and thus good diversification over a wide variety of asset classes is an important issue.

These papers on option pricing are mostly the result of a class I attended in Winter Quarter of 1999 at Northwestern University, "Mathematical Models of Finance," taught by Professor Don Saari. I owe Don a great debt - what little understanding I've managed to develop of the basic mathematical ideas behind modern finance is due largely to this wonderful teacher and mathematician. Don actually went through the full proof of the Black-Scholes equation in all its gory detail (it took all quarter). I still have his lecture notes, and it might be fun (albeit painful) to write them up here some day in my own words. The bulk of the proof involves solving the heat equation, though, which really isn't all that interesting (not even to me).

January 12, 1999. Last updated November 3, 2011.

How options are priced when the underlying asset has only two possible future states. Studying these trivial options helps develop insight into how real options and the Black-Scholes equation work. We learn about arbitrage, the law of one price, hedging, risk-aversion, and risk-neutral valuation.

March 7, 1999. Last updated November 3, 2011.

Just remember "stock + put = bond + call".

February 9, 1999. Last updated November 3, 2011.

In which we jump into our tardis and travel to a strange alternate universe where investors are risk-neutral instead of risk-averse. Surprisingly, in this universe the Black-Scholes equation for European option pricing is exactly the same as in our own universe. It's also ridiculously easy to derive. All we need is regular calculus - one simple evaluation of a definite integral derives the equation. We don't need any of the fancy machinery of stochastic calculus, arbitrage argument tricks, Ito's lemma, the heat equation, or parabolic differential equations that we use to derive Black-Scholes in our own universe.

February 17, 1999. Last updated November 3, 2011.

Some nicely drawn graphs of the Black-Scholes equation (thanks to Microsoft Excel). The main graph is a surface graph for a European call option on the S&P 500 stock market index, with strike price E = $100, the asset price S varying from $0 to $200, and the time to expiration t varying from 0 to 5 years. This kind of graph helps the beginner start to get a feeling for how the equation works. For people working through the derivation, the graph also helps visualize the boundary value conditions for the partial differential equation. The table of data values for the graph is included, with annotations and intermediate calculations shown in an attempt to develop some intuition for at least some of the parts of the equation and how they fit together. Not a complete triumph, but a start.

For people who have Excel (Mac or PC version), the spreadsheet is also available. Try changing the volatility parameter sigma and the risk-free rate parameter r and see how the surface graph changes. This helps develop a little bit more insight into the equation (and every little bit helps).

It would be nice to add more graphs, and maybe even play with some animation using Visual Basic scripts. Maybe some day I'll do this (and then again, maybe not).

Three Proofs that TSM is Efficient

April 4, 2004. Last updated November 3, 2011.

Many people do not understand why the cap-weighted total US stock market (TSM) plays such a central role in financial economics. They believe that TSM is just one of many possible US stock portfolios, with no good reason to believe that it is special or superior to other kinds of US stock portfolios. They often present alternatives which they claim offer a higher expected return than TSM with less risk. In technical terms, these alternatives are "more efficient" than TSM. We give three proofs that, under three different assumptions, TSM is efficient, in the sense that no other US stock portfolio can be more efficient than TSM (have lower risk and higher expected return). The three assumptions are:

If any one of these assumptions is true, TSM must be efficient. If any other US stock portfolio has a higher expected return than TSM with lower risk, we must reject all three of the assumptions.

- The Efficient Market Hypothesis.
- The Capital Asset Pricing Model.
- The Fama-French Three Factor Pricing Model.

An Exact Equation for the Median Return

April 23, 2004. Last updated November 3, 2011.

A short derivation of the exact equation for computing the median or "annualized" return of a lognormal random walk asset, given the simply compounded average or "expected" return and standard deviation of the simply compounded returns.

These articles were written for a more popular audience. They do not contain any hard math. They were written as attempts to explain my ideas to the nice folks on the Vanguard Diehards Forum at The Morningstar Web Site.

### Software

### Revision History

April 1, 2000. Last updated December 22, 2012.

A collection of arguments disputing the ubiquitous popular opinion that the risk of investing in volatile assets like stocks decreases as one's time horizon increases. This is an attempt to make some sense out of a profoundly important and deep issue which all investors should take seriously.

We discuss the fallacy of time diversification, the utility theory argument, the inadequacy of using probability of shortfall as a risk measure, an argument based on the theory of option pricing, and the impact of human capital on the problem.

January 9, 2002. Last updated November 3, 2011.

Financial academics and other experts often recommend that investors should use index mutual funds to invest in entire markets. This advice is common with both stocks and bonds and with both domestic and international investing.

Why do we so often hear this advice? When is it appropriate for an individual investor to not follow it or to use it as a starting point but deviate from it? These are the questions we address in this article.

Mean Reversion, Forecasting and Market Timing

December 1, 2005. Last updated November 3, 2011.

We develop a simple coin tossing game to explore the notions of reversion to mean ("RTM"), forecasting market returns, and market timing strategies. We mention recent research that casts doubts on both RTM and the ability to forecast returns.

February 1, 2005.

Version 2.0.A Java program that goes along with the Financial Planning Using Random Walks paper.

The program uses Monte Carlo simulations to implement the enhanced random walk model developed in the paper. It can graph single random walk simulations and ending value density and cumulative density functions. The cumulative density graphs are the ones that are most useful. The paper is the user manual for the program.

Here's a screen shot.

The program requires Java 1.4 or later. Apple Macintosh users running Mac OS X 10.3 or later already have Java 1.4. Mac users running Mac OS X 10.2 or later who do not already have Java 1.4 may get it by running Software Update. Microsoft Windows users who do not already have Java 1.4 can get it at Sun's web site. The program also works on Linux systems.

The program requires a screen size of at least 1024 by 768 pixels.

Copyright Notices and License StatementsThese are the copyright notices and license statements for Random Walker. Do not use the software if you do not accept the terms of the licenses described below.

Copyright © 2005, Northwestern University.

Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. Northwestern University makes no representations about the suitability of this software for any purpose. It is provided "as is" without expressed or implied warranty.

The program uses the Colt Distribution, which has the following copyright notice and license: Copyright © 1999 CERN - European Organization for Nuclear Research. Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. CERN makes no representations about the suitability of this software for any purpose. It is provided "as is" without expressed or implied warranty.

March 28, 2005.

Version 1.0.A Java program that goes along with the Portfolio Optimization: Part 2 - Constrained Portfolios paper.

The program implements the critical-line algorithm to solve portfolio optimization problems with optional low and high bound asset constraints. The paper is the user manual for the program.

Here's screen shot 1 and screen shot 2.

The program requires Java 1.4 or later. Apple Macintosh users running Mac OS X 10.3 or later already have Java 1.4. Mac users running Mac OS X 10.2 or later who do not already have Java 1.4 may get it by running Software Update. Microsoft Windows users who do not already have Java 1.4 can get it at Sun's web site. The program also works on Linux systems.

The program requires a screen size of at least 1024 by 768 pixels.

Copyright Notices and License StatementsThese are the copyright notices and license statements for PortOpt. Do not use the software if you do not accept the terms of the licenses described below.

Copyright © 2005, Northwestern University.

Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. Northwestern University makes no representations about the suitability of this software for any purpose. It is provided "as is" without expressed or implied warranty.

The program uses the Colt Distribution, which has the following copyright notice and license: Copyright © 1999 CERN - European Organization for Nuclear Research. Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. CERN makes no representations about the suitability of this software for any purpose. It is provided "as is" without expressed or implied warranty.

December 1, 2005

I added the new article Mean Reversion, Forecasting and Market Timing.

March 28, 2005

I added the new paper Portfolio Optimization: Part 2 - Constrained Portfolios and its companion program PortOpt.

February 1, 2005

I converted the Random Walker program from Macintosh C to cross-platform Java.

January 28, 2005

A major overhaul. This main web page has been cleaned up and reorganized.

I converted all of the math papers to LaTeX, the greatest invention for doing mathematics since Arabic numerals. Goodbye and good riddance to Microsoft Word! This took two weeks, but was well worth the effort. I took the opportunity to clean up quite a bit of verbiage as I went through all the papers.

The new papers Probability Review and Linear Algebra Review are not really new - they used to be sections in the Portfolio Optimization: Part 1 paper.

The paper Random Walks is partly new and partly old. The mathematical material at the beginning is new. The S&P 500 example used to be part of the Financial Planning Using Random Walks paper (which has a new title and has undergone a major reorganization).

Up to my home page.