In the video linked below, the Insurance Information Institute imagines what the world would be like without insurance. Spoiler alert: such a world would be a dystopian nightmare – it would be unsafe, much poorer, and significantly less innovative and resilient.

# How government policy exacerbates hurricane-related damage

One year ago this coming Sunday, the article cited below was the cover story for the 9/2/17 issue of *The Economist*. The points raised by this article (regarding the “moral hazard” associated with mispriced/subsidized insurance coupled with misguided NFIP claims policies) are (unfortunately) as valid today as they were back then.

Quoting from this article,

“Underpricing (of flood insurance) encourages the building of new houses and discourages existing owners from renovating or moving out. According to the Federal Emergency Management Agency, houses that repeatedly flood account for 1% of NFIP’s properties but 25-30% of its claims. Five states, Texas among them, have more than 10,000 such households and, nationwide, their number has been going up by around 5,000 each year. Insurance is meant to provide a signal about risk; in this case, it stifles it.”

As if global warming were not enough of a threat, poor planning and unwise subsidies make floods worse.

# Things That Make You Go Hmmm…

Financial historian John Stuart Gordon’s essay in today’s Wall Street Journal provides some particularly fascinating examples of rare events from the 19th, 20th, and 21st centuries!

Odds are these historical coincidences will strike you as unlikely.

# Problem Set 2 helpful hints

Problem Set 2 is now available from the course website at http://fin4335.garven.com/fall2018/ps2.pdf; its due date is Tuesday, September 4.

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 returns and standard deviations for 2-asset portfolios; see pp. 17-23 of the http://fin4335.garven.com/fall2018/lecture3.pdf lecture note for coverage of this topic. 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 tomorrow’s class meeting to these and related topics.

# 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 a particularly important risk management topic; i.e., 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.”

**Postscript:** The scene above depicts the early days of Lloyd’s Coffee House in London, England. According to Wikipedia, Lloyd’s Coffee House was opened by Edward Lloyd in 1686 and quickly became “… a popular place for sailors, merchants and shipowners, and Lloyd catered to them with reliable shipping news. The shipping industry community frequented the place to discuss maritime insurance, shipbroking and foreign trade. The dealing that took place led to the establishment of the insurance market Lloyd’s of London…”

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

Following up on the previous 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:

[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 77 students are enrolled in two sections of Finance 4335 this semester, this implies that the probability that two Finance 4335 students share the same birthday is roughly *p*’(77) = 1 – (364/365)*^*^{[77(76)/2] }= 99.97%,

# Statistical Independence

During today’s Finance 4335 class meeting, I introduced the concept of *statistical independence*. On Thursday, much of our class discussion will focus on the implications of statistical independence for probability distributions such as the binomial and normal distributions which we will rely upon throughout the semester.

Whenever risks are statistically independent of each other, this implies that they are uncorrelated; i.e., random variations in one variable are not meaningfully related to random variations in another. For example, auto accident risks are largely uncorrelated random variables; just because I happen to get into a car accident, this does not make it any more likely that you will suffer a similar fate (that is unless we happen to run into each other!). Another example of statistical independence is a sequence of coin tosses. Just because a coin toss comes up “heads,” this does not make it any more likely that subsequent coin tosses will also come up “heads.”

Computationally, the joint probability that we both get into car accidents or heads comes up on two consecutive tosses of a coin is equal to the product of the two event probabilities. Suppose your probability of getting into an auto accident during the coming year is 1%, whereas my probability is 2%. Then the likelihood that we both get into auto accidents during the coming year is .01 x .02 = .0002, or .02% (1/50th of 1 percent). Similarly, when tossing a “fair” coin, the probability of observing two “heads” in a row is .5 x .5 = 25%. The probability rule which emerges from these examples can be generalized as follows:

Suppose X

and X_{i}_{j}are uncorrelated random variables with probabilities p_{i}and p_{j}respectively. Then thejointprobability that both X_{i}andXoccur is equal to p_{j }_{i}p_{j}.

# Solutions for Problem Set 1…

… are now available at http://fin4335.garven.com/fall2018/ps1solutions.pdf.

# Problem Set 1 Hint…

Problem Set 1 is due at the beginning of class tomorrow. Here is a hint for solving the 4th question on this problem set.

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 the total cost of operating a hospital; a sure-fire way to minimize total cost would be to not even have a hospital in the first place. Indeed, if you were to differentiate the total cost function given by C = 4,700,000 + 0.00013X2 with respect to X, this is what the math would tell you.

In part “a” of the 4th question, you are asked to “derive” a formula for the relationship between cost per patient-day and the number of patient days; in other words, what you are interested in determining is what is the most cost-efficient way to scale a hospital facility such that the cost per patient-day is minimized. Once you obtain that equation, then you’ll be able to answer the question concerning optimal hospital size.