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

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.”

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 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.”

The Birthday Paradox: an interesting probability problem involving “statistically independent” events

Following up on last week’s blog posting  entitled “Statistical Independence,” consider the so-called “Birthday Paradox”. The Birthday Paradox pertains to the probability that in a set of randomly chosen people, some pair of them will have the same birthday. Counter-intuitively, in a group of 23 randomly chosen people, there is slightly more than a 50% probability that some pair of them will both have been born on the same day.

To compute the probability that two people in a group of n people have the same birthday, we disregard variations in the distribution, such as leap years, twins, seasonal or weekday variations, and assume that the 365 possible birthdays are equally likely.[1] Thus, we assume that birth dates are statistically independent events. Consequently, the probability of two randomly chosen people not sharing the same birthday is 364/365. According to the combinatorial equation, the number of unique pairs in a group of n people is n!/2!(n-2)! = n(n-1)/2. Assuming a uniform distribution (i.e., that all dates are equally probable), this means that the probability that no pair in a group of n people shares the same birthday is equal to p(n) = (364/365)^[n(n-1)/2]. The event of at least two of the n persons having the same birthday is complementary to all n birthdays being different. Therefore, its probability is p’(n) = 1 – (364/365)^[n(n-1)/2].

Given the assumptions listed in the previous paragraph, suppose that we are interested in determining how many randomly chosen people are needed in order for there to be a 50% probability that at least two persons share the same birthday. In other words, we are interested in finding the value of n which causes p(n) to equal 0.50. Therefore, 0.50 = (364/365)^[n(n-1)/2]; taking natural logs of both sides and rearranging, we obtain (ln 0.50)/(ln 364/365) = n(n-1)/2. Solving for n, we obtain 505.304 = n(n -1); therefore, n is approximately equal to 23.[2]

The following graph illustrates how the probability that a pair of people share the same birthday varies as the number of people in the sample increases:

New Picture (1)

[1] It is worthwhile noting that real-life birthday distributions are not uniform since not all dates are equally likely. For example, in the northern hemisphere, many children are born in the summer, especially during the months of August and September. In the United States, many children are conceived around the holidays of Christmas and New Year’s Day. Also, because hospitals rarely schedule C-sections and induced labor on the weekend, more Americans are born on Mondays and Tuesdays than on weekends; where many of the people share a birth year (e.g., a class in a school), this creates a tendency toward particular dates. Both of these factors tend to increase the chance of identical birth dates, since a denser subset has more possible pairs (in the extreme case when everyone was born on three days of the week, there would obviously be many identical birthdays!).

[2]Note that since 33 students are enrolled in Finance 4335 this semester, this implies that the probability that two Finance 4335 students share the same birthday is roughly p’(33) = 1 – (364/365)^[33(32)/2] = 76.5%.

Erratum…

Thanks to everyone who quickly corrected my errors in posting grades for the first time in Finance 4335.  This was caused by a careless cutting an pasting on my part.  The attendance grades as posted are now correct.  I still need to get the second problem set and third quiz (both of which were collected today at the beginning of class) graded and posted; I am sure this will occur sometime next weekend.

Quote for the day

There are very few things which we know, which are not capable of being reduced to a mathematical reasoning. And when they cannot, it’s a sign our knowledge of them is very small and confused. Where a mathematical reasoning can be had, it’s as great a folly to make use of any other, as to grope for a thing in the dark, when you have a candle standing by you.

—Of the Laws of Chance, Preface (1692)
John Arbuthnot (1667–1735)

It Has Been a Near-Perfect Investing Environment. But It May End Soon.

As this article from today’s WSJ points out, the “near-perfect” environment is in reference to a two decade-long financial market anomaly (dating back to the late 1990s) in which stock and bond  have tended to move in opposite directions.  Thus, investors have been able to (quite effectively) hedge risk by owning both asset classes.
For two decades, government bonds have provided what amounts to free insurance against stock-market struggles. But that’s a historical anomaly.

Stocks Weren’t Made for Social Climbing

Superb WSJ op-ed by (former hedge fund manager turned author) Andy Kessler about the corporate social responsibility “gospel” and the importance of profit; Kessler’s essay is essentially an homage to Milton Friedman’s famous 1970 New York Times Magazine article entitled “The Social Responsibility of Business Is to Increase Its Profits.”

Profits are the proper gauge of a company’s value to consumers—and to society.

Problem Set 2 helpful hints

Problem Set 2 is now available from the course website at http://fin4335.garven.com/spring2018/ps2.pdf; its due date is Thursday, January 25.

Problem Set 2 consists of two problems.  The first problem requires calculating expected value, standard deviation, and correlation, and using this information as inputs into determining expected return and standard deviation for 2-asset portfolios.  We covered these concepts during last Thursday’s statistics tutorial; also see pp. 10-20 of the http://fin4335.garven.com/spring2018/lecture3.pdf lecture note.  The second problem involves using the standard normal probability distribution to calculate probabilities of earning various levels of return by investing in risky securities and portfolios.  We will devote next Tuesday’s class meeting to this and related topics.

Plans for next week’s Finance 4335 class meetings, along with a preview of future topics

This coming Tuesday, we will complete the probability and statistics tutorial by studying the binomial and normal probability distributions.

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 15). Starting on Thursday, we will begin our “dive” into 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 our attention upon variance as our risk measure. Most basic finance models (e.g., portfolio theory and the capital asset pricing model, or CAPM) 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 to which 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 us with a very 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, we’ll move on to other topics including demand for insurance, asymmetric information, portfolio theory, capital market theory, option pricing theory, and corporate risk management.

Finance 4335