I. Handouts and Excel Spreadsheets for Differential
Equations:
Several days in the differential equations syllabus require the notes and
spreadsheets provided here.
- Days 27 and 28:
Convolutions and the Impulse Response Function:
notes
If all that the Laplace transform did was to provide a new approach to problems that could
already be solved by other more straightforward methods, it would have been tossed into the
dustbin of history long ago. In these notes, we introduce a peculiar mathematical operator called
the convolution, and use it to discover one of the triumphs of the Laplace transform.
-
Day 33: The
Lorenz Equations: notes, spreadsheet
Ever wonder why weather forecasters have such a hard time with predictions more than a few days into the future? They have sophisticated mathematical models, giant computers, data from weather stations, satellites and balloons,
yet can't reliably say whether it's going to be rainy or sunny next weekend. Edward Lorenz and his 50 year old,
740 pound computer might just have an answer for you....
(click the screenshot to try out the spreadsheet)
-
Day 41:
The Lanchester Equations:
notes
Wouldn't it be nice if, before a military encounter,
you could reliably predict who would emerge victorious and how many casualties the victor would
take? In the thick of World War I, F.W. Lanchester, a British engineer in the Royal Air Force, used a
little bit of mathematics to tackle this challenge.
-
Day 48: Intro to the
Heat Equation: notes, spreadsheet
If you drop a small amount of dye into a glass of water, that dye will
slowly spread until the entire volume of water is a uniform hue.
The heat equation is a partial differential equation that
provides a model for the change in the density of the dye at any
position as time goes by. Separation of variables can be used to find
solutions of the heat equation, and we especially note the solutions with sines and cosines
in the spatial variable that can be used as "building blocks" for more general initial
profiles.
(click the screenshot to try out the spreadsheet)
II. Java Tools from Mathematical Finance:
These two tools illustrate some of the major mathematical themes in Modern Portfolio Theory.
- 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 RN. 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 RN. 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)
- 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)
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