Excel Spreadsheets in Core Mathematics:

Many mathematical ideas -- especially recursive methods like Newton's method or Euler's Method -- can be implemented using a simple spreadsheet.  This requires no specialized knowledge of programming languages, and is something that many students are already familiar with.

1. Euler's Method in Higher Dimensions:
   
   Euler's Method is a simple numerical scheme for approximating the solution of a differential equation.  In a sense, using Euler's Method to approximate the solution of a differential equation is like driving a car through an unknown town.  You ask a local, "which way should I go?", the local quietly points his finger down the road, and so you drive in that direction for a short while.  Then you stop, roll down the window and ask another local for further instruction.
   
   Here's a handout (euler.doc) and a sample spreadsheet (euler.xls) to get you started....

   

   (click the screenshot to try out the spreadsheet)

Java Tools from Mathematical Finance:

1. Visual Fama-French Tool:
   
   Although the average day trader spends his hours flying in and out of individual stock positions, much academic research (see Sharpe or Fama and French) has suggested that the large majority of a portfolio's returns can be attributed to its exposure to one or more identifiable risk factors.  Sharpe's single factor (beta) represents the degree to which a security (or a portfolio) moves with the overall market.  Fama and French include two additional factors in their analysis -- representing the unique risks of small companies (smb) and value companies (hml).
   With just a touch of matrix algebra, we can see how any portfolio stacks up against these factors.  Using the historical data, we can compute an annual value for each of the risk factors identified above:
      1) the market return minus the risk-free rate
      2) the return of small stocks minus the return of large
      3) the return of stocks with high BTM ratios minus the return of stocks with low
   Computing these three factors for N years gives us three vectors in R^N.  We then include a fourth vector, made up of all ones to model the annual excess return that all of our machinations beyond this factor exposure (for example, skill, luck and costs)  has gained us.  These four vectors span a 4-dimensional subspace of R^N. To understand the returns of our own portfolio, we simply project the vector made up of our portfolio returns onto this space!  How does your portfolio measure up?

(click the screenshot to try out the applet)

2. Visual Efficient-Frontier Tool:
   
   In general, the amount of return you should expect from your portfolio is related to the amount of risk you take. This applet allows you to combine a diverse family of five assets (using return data from 1972-2007) into a portfolio.  By combining these different assets appropriately, we can find portfolios that give us the most (return) bang for our (risk) buck.
   In this applet, "risk" is represented by logarithmic deviation -- the log scaling allows a balance between a portfolio doubling and a portfolio halving.  "return" is represented by the annualized return over the entire time interval.  In Modern Portfolio Theory, a portfolio is on the efficient frontier  if it provides the maximum return for a fixed amount of risk.  Can you find the portfolios on the (historical) efficient frontier in this applet?
   

(click the screenshot to try out the applet)