In his video lesson entitled “Visualizing Taylor polynomial approximations“, Sal Kahn essentially replicates the tail end of last Thursday’s Finance 4335 class meeting in which we approximated y = eˣ with a Taylor polynomial centered at x=0. 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 one can approximate functions using Taylor polynomials, and the accuracy of the approximation increases as the order of the polynomial increases (see pp. 19-25 in my Mathematics Tutorial lecture note if you wish to review what we did in class on Thursday).
Next week in Finance 4335 will be devoted to tutorials on probability and statistics. These tools are critically important in order to evaluate risk and develop appropriate risk management strategies for individuals and firms alike. Next Tuesday’s class meeting will be devoted to introducing discrete and continuous probability distributions, calculating parameters such as expected value, variance, standard deviation, covariance and correlation, and applying these concepts to measuring expected returns and risks for portfolios consisting of risky assets. Next Thursday will provide a deeper dive into discrete and continuous probability distributions, in which the binomial and normal distributions are showcased.
While I have your attention, let me briefly explain what the main “theme” will initially be in Finance 4335 (up to the first midterm exam, which is scheduled for Thursday, February 21). Starting on Tuesday, January 29, we will begin our discussion of decision theory. Decision theory addresses decision making under risk and uncertainty, and not surprisingly, risk management lies at the very heart of decision theory. Initially, we’ll focus attention on variance as our risk measure. Most basic finance models (e.g., portfolio theory, the capital asset pricing model (CAPM), and option pricing theory) implicitly or explicitly assume that risk = variance. We’ll learn that while this is not necessarily an unreasonable assumption, circumstances can arise where it is not an appropriate assumption. Furthermore, since individuals and firms are typically exposed to multiple sources of risk, we need to take into consideration the portfolio effects of risk. To the extent that risks are not perfectly positively correlated, this 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 particularly 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 quite a bit of ambiguity when we make decisions without complete information, but 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, the remainder 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.
This week’s Intelligent Investor column in the Wall Street Journal presents an homage to the memory of Jack Bogle, the founder of Vanguard Group. Mr. Bogle passed away this past Wednesday at the age of 89, and as the inventor of index investing, he is arguably one of the most important public figures in the practice of finance of the past 50 years. Burton Malkiel’s WSJ op-ed in today’s paper entitled “The Secrets of Jack Bogle’s Investment Success” is also a must read!
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 of 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.”
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 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.”
Problem Set 1 is due at the beginning of class on Tuesday, January 22. Here is a hint for solving the 4th question on problem set 1.
The objective is to determine how big a hospital must be so that the cost per patient-day is minimized. We are not interested in minimizing total cost; if this were the case, there would be no hospital because marginal costs are positive, which implies that total cost is positively related to the number of patient-days.
The cost equation C = 4,700,000 + 0.00013X2 tells you the total cost as a function of the number of patient-days. This is why you are asked in part “a” of the 4th question to derive a formula for the relationship between cost per patient-day and the number of patient days. Once you have that equation, then that is what you minimize, and you’ll be able to answer the question concerning optimal hospital size.
It just so happens that Hoover Senior Fellow (and former Univ. of Chicago Finance professor) John Cochrane posted an article yesterday entitled “Volatility, now the whole thing” which builds and expands upon today’s implied volatility topic in Finance 4335. Cochrane’s article provides a broader framework for thinking critically about the implications of volatility for future states of the overall economy. This article is well worth everyone’s time and attention, so I highly encourage y’all to read it!
There is a section in the assigned “Optimization” reading due Thursday, 1/17 on pp. 74-76 entitled “Lagrangian Multipliers” which (as noted in footnote 9 of that reading) may be skipped without loss of continuity. The primary purpose of this chapter is to re-acquaint students with basic calculus and how to use the calculus to solve so-called optimization problems. Since the course only requires solving unconstrained optimization problems, there’s no need for Lagrangian multipliers.
Besides reading the articles entitled “Optimization” and “How long does it take to double (triple/quadruple/n-tuple) your money?” in preparation for this coming Thursday’s meeting of Finance 4335, make sure that you fill out and email the student information form as a file attachment to firstname.lastname@example.org prior to the beginning of tomorrow’s class. As I explained during today’s class meeting, this assignment counts as a problem set, and your grade is 100 if you turn this assignment in on time (i.e., sometime prior to tomorrow’s class meeting) and 0 otherwise.
At any given point in time during this semester, you can ensure that you are on track with Finance 4335 assignments by monitoring due dates which are published on the course website. See http://fin4335.garven.com/readings/ for due dates pertaining to reading assignments, and http://fin4335.garven.com/problem-sets/ for due dates pertaining to problem sets. Also, keep in mind that short quizzes will be administered in class on each of the dates indicated for required readings. As a case in point, since the required readings entitled “Optimization” and ” How long does it take to double (triple/quadruple/n-tuple) your money?” are listed for Thursday, January 17, this means that a quiz based upon these readings will be given in class on that day.
Important assignments due on the second class meeting of Finance 4335 (scheduled for Thursday, January 17) include: 1) filling out and emailing the student information form as a file attachment to email@example.com, 2) subscribing to the Wall Street Journal, and 3) subscribing to the course blog. A completed student information form is graded as a problem set and receives 100 points; if you fail to turn in a student information form, then you will receive a 0 for this “problem set”. Furthermore, tasks 2 and 3 listed above count toward your class participation grade in Finance 4335.
Regarding the student information form, I prefer that you complete this form (by either typing or writing) and email it to firstname.lastname@example.org prior to the beginning of class on Thursday, January 17. However, if you prefer, you may turn in a hard copy instead at the beginning of class on that day.
Besides going over the course syllabus during the first day of class on Tuesday, January 15, we will also discuss a particularly important “real world” example of financial risk. Specifically, we will look at the relationship between stock market returns (as indicated by daily percentage changes in the SP500 stock market index) and stock market volatility (as indicated by daily percentage changes in the CBOE Volatility Index (VIX)):
As indicated by this graph from page 21 of the lecture note for the first day of class, daily percentage changes on closing prices for VIX and the SP500 are strongly negatively correlated. In the graph above, the y-axis variable is the daily return on the SP500, whereas the x-axis variable is the daily return on the VIX. The blue points represent 7,311 daily observations on these two variables, spanning the time period from January 2, 1990 through January 7, 2019. 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 (-0.1139) indicates that on average, daily VIX returns during this time period were inversely related to the daily return on the SP500; i.e., when volatility as measured by VIX went down (up), then the stock market return as indicated by SP500 typically went up (down). Nearly half of the variation in the stock market return during this time period (specifically, 48.73%) can be statistically “explained” by changes in volatility, and the correlation between and comes out to -0.696. While a correlation of -0.698 does not imply that and will always move in opposite directions, it does indicate that this will be the case more often than not. Indeed, closing daily returns on and during this period moved inversely 78.4% of the time.
You can 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.