### Calculus, Probability and Statistics, and a preview of future topics in Finance 4335

Probability and statistics, along with the basic calculus principles covered last Thursday, are foundational for the theory of pricing and managing risk with financial derivatives, which is what this course is all about. During yesterday’s class meeting, we introduced discrete and continuous probability distributions, calculated parameters such as expected value, variance, standard deviation, covariance, and correlation, and applied these concepts to measure expected returns and risks for portfolios comprising risky assets. During tomorrow’s class meeting, we will take a deeper dive into discrete and continuous probability distributions, in which the binomial and normal distributions will be showcased.

While I have your attention, let me briefly explain what the main “theme” will initially be in Finance 4335.  Starting on Tuesday, February 2, we will begin our discussion of decision theory. Decision theory addresses decision-making under risk and uncertainty, which at the very heart of risk management. Initially, we’ll focus attention on variance as our risk measure. Most of the basic finance theories, including portfolio, capital market, and option pricing theories, define risk as variance. We’ll learn that while this is not necessarily an unreasonable assumption, circumstances may arise where it is not an appropriate assumption. Since individuals and firms encounter multiple sources of risk, we also need to take into consideration the portfolio effects of risk. Portfolio theory implies that risks often “manage” themselves by canceling each other out. Thus the risk of a portfolio is typically less than the sum of the individual risks which comprise the portfolio.

The decision theory provides a useful framework for thinking about concepts such as risk aversion and risk tolerance. The calculus comes in handy by providing an analytic framework for determining how much risk to retain and how much risk to transfer to others. Such decisions occur regularly in daily life, encompassing practical problems such as deciding how to allocate assets in a 401-K or IRA account, determining the extent to which one insures health, life, and property risks, whether to work for a startup or an established business and so forth. There’s also ambiguity when we have incomplete information about risk. This course will at least help you think critically about costs, benefits, and trade-offs related to decision-making whenever you encounter risk and uncertainty.

After the first midterm (scheduled for Tuesday, February 23)., the rest of the semester will be devoted to various other risk management topics, including the demand for insurance, asymmetric information, portfolio theory, capital market theory, option pricing theory, and corporate risk management.

### Also featured as one of “50 Things That Made the Modern Economy”: The Index Fund

Besides insurance, Tim Harford also features the index fund in his “Fifty Things That Made the Modern Economy” radio and podcast series. This 9-minute long podcast lays out the history of the development of the index fund in particular and the evolution of so-called passive portfolio strategies in general. Much of the content of this podcast is sourced from Vanguard founder Jack Bogle’s September 2011 WSJ article entitled “How the Index Fund Was Born” (available at https://www.wsj.com/articles/SB10001424053111904583204576544681577401622). Here’s the description of this podcast:

“Warren Buffett is the world’s most successful investor. In a letter he wrote to his wife, advising her how to invest after he dies, he offers some clear advice: put almost everything into “a very low-cost S&P 500 index fund”. Index funds passively track the market as a whole by buying a little of everything, rather than trying to beat the market with clever stock picks – the kind of clever stock picks that Warren Buffett himself has been making for more than half a century. Index funds now seem completely natural. But as recently as 1976 they didn’t exist. And, as Tim Harford explains, they have become very important indeed – and not only to Mrs Buffett.”

Warren Buffett is one of the world’s great investors. His advice? Invest in an index fund

### Insurance featured as one of “50 Things That Made the Modern Economy”

From November 2016 through October 2017, Financial Times writer Tim Harford presented an economic history documentary radio and podcast series called 50 Things That Made the Modern Economy. This same information is available in book form under the title “Fifty Inventions That Shaped the Modern Economy“. While I recommend listening to the entire series of podcasts (as well as reading the book), I would like to call your attention to Mr. Harford’s episode on the topic of insurance, which I link below. This 9-minute long podcast lays out the history of the development of the various institutions which exist today for the sharing and trading of risk, including markets for financial derivatives as well as for insurance.

“Legally and culturally, there’s a clear distinction between gambling and insurance. Economically, the difference is not so easy to see. Both the gambler and the insurer agree that money will change hands depending on what transpires in some unknowable future. Today the biggest insurance market of all – financial derivatives – blurs the line between insuring and gambling more than ever. Tim Harford tells the story of insurance; an idea as old as gambling but one which is fundamental to the way the modern economy works.”

### Week 2 readings, quiz, and problem set

Here’s a friendly reminder that the following readings are due tomorrow:

1. The New Religion of Risk Management, by Peter Bernstein
2. Normal and standard normal distribution, by James R. Garven
3. Mean and Variance of a Two-Asset Portfolio, by James R. Garven

Keep in mind that Quiz 2, which is based on these readings, must be completed prior to the start of class tomorrow. I have also changed the due date for Problem Set 1 from tomorrow (Tuesday, January 26) to Thursday, January 28. You should consider all other due dates listed on Canvas and on the course website as pretty much set in stone for the remainder of the semester.

Going forward, I will typically not post reminders like this concerning Finance 4335 assignment deadlines; however, you’ll be "good to go" in Finance 4335 if you faithfully follow the guidelines listed in my "How to know whether you are on track with Finance 4335 assignments" posting.

### Barron’s in Education 30-minute live event: How Student Loan Policy May Change Under A Biden Administration

This Wednesday, 1/27/21, at 11 AM CT: How Student Loan Policy May Change Under A Biden Administration. Join MarketWatch reporter Jillian Berman and Seth Frotman, Founder and Executive Director of the Student Borrower Protection Center for a discussion on the student loan market, the challenges borrowers are facing, and the policy changes we could see under the Biden administration.

Click here to register for Barron’s Live.

### Rules for calculating (math) derivatives

Here’s a particularly useful list of rules for calculating (math) derivatives:

### On the ancient origin of the word “algorithm”

The January 26th 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 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.

### 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 4335 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.

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

At any point in time 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 at 1:30 p.m. CT the day before the reading assignment is due.