Category: Math and Statistics

Posted on

Z Table Extra Credit Assignment (due at the start of class on Tuesday, January 30)

Here’s an extra credit opportunity for Finance 4366. Working on your own (i.e., this is not a group project; I will only give credit for spreadsheets that are uniquely your own), build your own “z” table in Excel (patterned after the table at http://fin4366.garven.com/stdnormal.pdf); the top row should have values ranging from 0.00 to 0.09, and the first column should have z values ranging from -3.0 to +3.0, in increments of 0.1).

Conveniently, Excel has the standard normal distribution function built right in; e.g., if you type “=normsdist(z)”, Excel returns the probability associated with whatever z value you provide. If you type “=normsdist(0)”, .5 is returned, since half of the area under the curve lies to the left of the expected value E(z) = 0. Similarly, if you type “=normsdist(1)”, then .8413 is returned because 84.13% of the area under the curve lies to the left of z = 1. Perhaps you recall from your QBA course that 68.26% of the area under the curve lies between z = -1; this “confidence interval” of +/- 1 standard deviation away from the mean (E(z)=0) is calculated in Excel with the following code: “=normsdist(1)-normsdist(-1)”, and so forth.

The grade you earn on this extra credit assignment will replace your lowest quiz grade; that is if your lowest quiz grade is lower than your extra credit grade. The deadline is the start of class on Tuesday, January 30.

You can turn your spreadsheet for this extra credit assignment in at the link labeled “Z Table Extra Credit Assignment” under the Assignment tab on Canvas.

Posted on

On the ancient origin of the word “algorithm”

The January 24th assigned reading entitled “The New Religion of Risk Management” (by Peter Bernstein, March-April 1996 issue of Harvard Business Review) provides a succinct synopsis of the same author’s 1996 book entitled “Against the Gods: The Remarkable Story of Risk“. Here’s a fascinating quote from page 33 of “Against the Gods” which explains the ancient origin of the word “algorithm”:

“The earliest known work in Arabic arithmetic was written by al­Khowarizmi, a mathematician who lived around 825, some four hun­dred years before Fibonacci. Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al­Khowarizmi” fast. That’s where we get the word “algo­rithm,” which means rules for computing.”

Note: The book cover shown above is a copy of a 1633 oil-on-canvas painting by the Dutch Golden Age painter Rembrandt van Rijn.
Posted on

Origin of the “Product Rule”, and Visualizing Taylor polynomial approximations

This blog entry provides a helpful follow-up for a couple of calculus-related topics that we covered during today’s Mathematics Tutorial.

  1. See page 12 of the above-referenced lecture note.  There, the equation for a parabola (y = {x^2}) appears, and the claim that \frac{{dy}}{{dx}} = 2x is corroborated by solving the following expression:
    In the 11-minute Khan Academy video at https://youtu.be/HEH_oKNLgUU, Sal Kahn takes on the solution of this problem in a very succinct and easy-to-comprehend fashion.
  2. In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn replicates the tail end of today’s Finance 4366 class meeting in which we approximated y = ex with a Taylor polynomial centered at x=0 (as also shown in pp. 18-23 of the Mathematics Tutorial lecture note).  Sal approximates y = ex with a Taylor polynomial centered at x=3 instead of x=0, but the same insight obtains in both cases, which is that the accuracy of Taylor polynomial approximations increases as the order of the polynomial increases.
Posted on

The 17 equations that changed the course of history (spoiler alert: we use 4 of these equations in Finance 4366!)

I especially like the fact that Ian Stewart includes the famous Black-Scholes equation (equation #17) on his list of the 17 equations that changed the course of history; Equations (2), (3), (7), and (17) play particularly important roles in Finance 4366!

The 17 equations that changed the course of history
From Ian Stewart’s book, these 17 math equations changed the course of human history.
Posted on

Calculus and Probability & Statistics recommendations…

Since most topics covered in Finance 4366 presuppose a fundamental understanding and facility with algebra, differential calculus, and probability & statistics, the second class meeting in the Spring 2024 semester will incorporate a mathematics tutorial. Subsequently, the third and fourth class meetings will delve into probability and statistics.

The Khan Academy is an exceptional online resource for revisiting or acquainting oneself with these concepts. Below are my recommendations for specific Khan Academy videos covering the following topics:

Lastly, if your algebra proficiency is somewhat rusty, I would also recommend exploring the Khan Academy’s algebra review for a comprehensive refresher.