… are available at http://fin4335.garven.com/fall2021/ps3solutions.pdf.

## Risk Aversion Class Problem Solutions

Here are the solutions for the Risk Aversion Class Problem that we worked on during our September 14 and September 16 class meetings.

## Quiz 4 reminder!

Quiz 4, which is available now and due tomorrow by 5 pm, is based on the following two required readings, both of which are available from the Finance 4335 website:

1. Expected Utility, Mean-Variance, and Stochastic Dominance, by James R. Garven

2. Modeling Risk Preferences Using Taylor Series Expansions of Utility Functions, by James R. Garven

## On the Determinants of Risk Aversion

This week, we completed the first two of a series of five Finance 4335 class meetings (scheduled for September 7-21) devoted to decision-making under risk and uncertainty. Our focus is on measuring risk, modeling consumer and investor risk preferences, and exploring implications for the pricing and management of risk. We focus especially on the concept of *risk aversion*. Other things equal, risk averse decision-makers prefer less risk to more risk. Risk aversion helps to explain some very basic facts of human behavior; e.g., why investors diversify, why consumers purchase insurance, etc.

Several years ago, *The Economist *published a particularly interesting article about various behavioral determinants of risk aversion, entitled “Risk off: Why some people are more cautious with their finances than others”. Here are some key takeaways from this (somewhat dated, but still quite timely) article:

- Economists have long known that people are risk averse, yet the willingness to run risks varies enormously among individuals and over time.
- Genetics explains a third of the difference in risk-taking; e.g., a Swedish study of twins finds that identical twins had “… a closer propensity to invest in shares” than fraternal ones.
- Upbringing, environment, and experience also matter; e.g., “… the educated and the rich are more daring financially. So are men, but apparently not for genetic reasons.”
- People’s financial history has a strong impact on their taste for risk; e.g., “… people who experienced high (low) returns on the stock market earlier in life were, years later, likelier to report a higher (lower) tolerance for risk, to own (not own) shares and to invest a bigger (smaller) slice of their assets in shares.”
- “Exposure to economic turmoil appears to dampen people’s appetite for risk irrespective of their personal financial losses.” Furthermore, low tolerance for risk is linked to past emotional trauma.

## Solutions for Problem Set 2…

… are available at http://fin4335.garven.com/fall2021/ps2solutions.pdf!

## Actuarially Fair Price of Insurance Policy

A Finance 4335 student asked me the following question during today’s office hours:

Q: “How do you find the actuarially fair price (premium) for an insurance policy?”

Here’s my answer to this question:

A: The actuarially fair price (premium) corresponds to the expected value of the insurance indemnity; the indemnity is the amount of coverage offered by an insurance policy. Under “full coverage”, 100% of the loss is indemnified, and in such a case, the actuarially fair premium is equal to the expected value of the loss distribution.

For what it’s worth, the concept of “actuarially fair” insurance prices/premiums, along with implications for the demand for insurance, is explained in two previously assigned readings (italics added for emphasis):,

- on page 4 of the Supply of Insurance reading (just prior to the section entitled “Example 2: Correlated Identically Distributed Losses), the following sentence appears, “A premium that is equal to the expected outcome is called an
*actuarially fair premium*”; - on page 30 of the Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2) reading, consider the following excerpt: “Ignoring transaction costs, an insurer charging a premium equal to expected loss would break even if it held a large portfolio of such policies. This premium could be called a fair premium or
*an actuarially fair premium*, denoting that the premium is equal to the expected value of loss (sometimes called the actuarial value of the policy). The term fair is not construed in a normative sense; rather it is simply a reference point”; and - on page 43 of Doherty, Chapter 2, in the first sentence of the first full paragraph: “We know from the Bernoulli principle that a risk averter will choose to fully insure at an actuarially fair premium.””

## Risk pooling class problem and solutions

The risk pooling problem that we worked on during yesterday’s meeting of Finance 4335 is available at http://fin4335.garven.com/fall2021/rpclassproblem.pdf; solutions are posted at http://fin4335.garven.com/fall2021/rpclassproblemsolutions.pdf.

## Coming up next in Finance 4335

On Tuesday, September 7, we will complete the Statistics Tutorial, Part 2 (specifically the topics that appear on pp. 11-23 of that lecture note), and move on to Decision Making under Risk and Uncertainty, part 1. Although I have kept the September 7 due date for the Supply of Insurance, Basic Economics: How Individuals Deal with Risk, and Introduction to Expected Utility and Risk Preferences assigned readings, I have moved the due date (because of the Labor Day holiday) for Quiz 3 from Tuesday, 9/7 to Thursday, 9/9. Also, Problem Set 2 will now be due on Thursday, 9/9 rather than Tuesday, 9/7.

## PDF Requirement for Finance 4335 assignments posted to Canvas

Whenever you post Finance 4335 assignments to Canvas, you are required to submit each and every assignment as a single file in the Adobe Portable Document (PDF) file format. For handwritten work, one must scan such work and output the scan to PDF.

I handed out a 2-page PDF scanning guide during class yesterday (also available here) which shows how one can accomplish this task easily using smartphone apps; specifically, the Adobe Scan app and the OneDrive app. Both apps are free, remarkably easy to use, and produce very high-quality scans. There may also be other apps that are functionally similar to Adobe Scan and OneDrive, and perhaps some of you use hardware-based scanning devices; any of these scanning options suffice as long as you can output and upload your work to a single PDF file.

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

During the statistics tutorial, we have discussed (among other things) the concept of *statistical independence* and focused attention on some important implications of statistical independence for the binomial probability distribution.

Here, I’d like to call everyone’s attention to an interesting (non-finance) probability problem related to statistical independence. Specifically, 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 these assumptions, 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 54 students are enrolled in Finance 4335 at Baylor University this semester, this implies that the probability that two Fall 2021 Finance 4335 students share the same birthday is *p*’(54) = 1 – (364/365)*^*^{[54(53)/2] }= 98.03%, although given footnote 1’s caveats, it’s likely that there may be several shared birthday pairs.