… are available at http://fin4335.garven.com/fall2022/ps1solutions.pdf.

### 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.”

### Visualizing Taylor polynomial approximations

On pp. 18-23 of the Mathematics Tutorial, I show how *y = **e*^{x} can be approximated with a Taylor polynomial centered at *x*=0 for values ranging from -2 to +2. In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn essentially replicates my work; the only difference between Sal’s numerical example and mine is that Sal approximates *y = **e*^{x} with a Taylor polynomial centered at *x*=3 instead of *x*=0. The important insight provided in both cases is that the accuracy of Taylor polynomial approximations increases as the order of the polynomial increases.

### Rules for calculating (math) derivatives

### How to know whether you are on track with Finance 4335 assignments

At any point in time throughout this semester, you can make sure that you are on track with Finance 4335 assignments by monitoring due dates on Canvas and on the course website. Links for future class meetings, quizzes, problem sets, and exams appear on the Canvas “To Do” list. Links for readings (along with their due dates) appear on http://fin4335.garven.com/readings/, and links for problem sets (along with their due dates) appear on http://fin4335.garven.com/problem-sets/. In the case of assigned readings, students are required to complete a short (10-minute) readings quiz prior to the start of class for each reading assignment due date; the window for completing this task begins 24 hours prior to the start of the class meeting for which the reading assignment is due.

### Friendly reminder about Finance 4335 plans for tomorrow (Thursday, 1/19)

### Assignments for Thursday, January 19 in Finance 4335

As indicated on the http://fin4335.garven.com/readings page of the course website, the readings assigned for the second day of class (Thursday, January 19) include

1. Calculating (Math) Derivatives, by James R. Garven

2. Optimization (chapter 2 from *Managerial Economics*, by W. Bruce Allen, Neil Doherty, Keith Weigelt, and Edwin Mansfield, 6th edition (2005))

3. How long does it take to double (triple/quadruple/n-tuple) your money?, by James R. Garven

Pages 74-76 of the “Optimization” reading (entitled “Lagrangian Multipliers”) may be skipped without loss of continuity. The primary purpose of this reading is to re-acquaint students with basic calculus and how to use calculus to solve (*unconstrained*) optimization (i.e., maximization and minimization) problems. Since none of our work in Finance 4335 requires solving *constrained* optimization problems, there’s no need (in this course, anyway) for Lagrangian multipliers.

Besides reading these three articles in preparation for the second day of class, Finance 4335 students are also required to complete Quiz 1 on Canvas, fill in the Student information survey, and sign up for a free *WSJ* student membership at https://wsj.com/ActivateBaylor.

### On the relationship between the S&P 500 and the CBOE Volatility Index (VIX)

Besides going over the course syllabus during the first day of class on Tuesday, January 17, we will also discuss a particularly important “real world” example of financial risk. Specifically, we will study the relationship between realized daily stock market returns (as measured by daily percentage changes in the SP500 stock market index) and changes in forward-looking investor expectations of stock market volatility (as indicated by daily percentage changes in the CBOE Volatility Index (VIX)):

As indicated by this graph (which also appears in the lecture note for the first day of class), daily percentage changes on closing prices for the SP500 (the y-axis variable) and for the VIX (the x-axis variable) are strongly negatively correlated with each other. The blue dots are based on 8,315 contemporaneous observations of daily returns for both variables, spanning the 33-year period of time starting on January 2, 1990 and ending on December 30, 2022. When we fit a regression line through this scatter diagram, we obtain the following equation:

,

where corresponds to the daily return on the SP500 index and corresponds to the daily return on the VIX index. The slope of this line (-0.1147) indicates that *on average*, daily closing SP500 returns are *inversely related* to daily closing VIX returns. Furthermore, nearly half of the variation in the stock market return during this time period (specifically, 48.87%) can be statistically “explained” by changes in volatility, and the correlation between and came out to -0.70. While a correlation of -0.70 does not imply that daily closing values for and always move in opposite directions, it *does* suggest that this will be the case more often than not. Indeed, closing daily values recorded for and during this period moved inversely 78.59% of the time.

You can also see how the relationship between the SP500 and VIX evolves prospectively by entering http://finance.yahoo.com/quotes/^GSPC,^VIX into your web browser’s address field.

### Calculus and Probability & Statistics recommendations…

Since many of the topics covered in Finance 4335 require a basic knowledge and comfort level with algebra, differential calculus, and probability & statistics, the second class meeting will include a mathematics tutorial, and the third and fourth class meetings will cover probability & statistics. I know of no better online resource for brushing up on (or learning for the first time) these topics than the Khan Academy.

So here are my suggestions for Khan Academy videos that cover these topics (unless otherwise noted, all sections included in the links which follow are recommended):

- Algebra: Intro to the Binomial Theorem, Pascal’s Triangle and Binomial Expansion
- Calculus: Taking derivatives, Optimization (profit maximization) with calculus, Visualizing Taylor Series for e^x
- Probability and statistics: Basic probability, Compound, independent events, Permutations, Combinations, probability using combinatorics, Random variables and probability distributions, Binomial distribution, Law of Large Numbers, and Normal Distribution.

Finally, if your algebra skills are a bit on the rusty side, I would also recommend checking out the Khan Academy’s review of algebra.

### Course Requirement: Email subscription to the Risk Management Course Blog (instructions given here)

A course blog has been established for Finance 4335 at the address http://risk.garven.com; it is also linked from the “Course Blog” button located on the course website. This resource provides a convenient means for Dr. Garven to distribute important announcements outside of class. Topics covered on the course blog typically include things like changes in the course schedule, clarifications, and hints concerning problem sets, information about upcoming exams, announcements concerning extra credit opportunities, and short blurbs showing how current events relate to many of the topics covered in Finance 4335.

All students enrolled in Finance 4335 are *required* to subscribe to the course blog via email.

**Email Subscription Instructions:**