Q&A from earlier today about the Midterm 1 exam in Finance 4335

Student Query:  Hello Dr. Garven, I had a quick question regarding the upcoming exam. Will questions only be pulled from information pertaining to problem sets 3 & 4? Or will there be questions about the math and stat review portion as well?

Dr. Garven’s Response:  Students will find the study guide to be very helpful in preparing for Tuesday’s midterm exam in Finance 4335. Also, see http://risk.garven.com/2022/09/21/midterm-exam-1-formula-sheet-and-helpful-hints/ for some helpful hints.  The exam covers topics from problem sets 2-4 and the various class problems. As I mentioned during Thursday’s exam review session, I recommend reviewing the various PDFs linked on the Problem Set Solutions page.  Calculus principles also play an important role in the theory of risk aversion (e.g., the math behind the Arrow-Pratt framework helps explain differences in the degree to which decision-makers are risk averse), and the application of Taylor series math to expected utility indicates that variance is an inadequate risk measure if risks are skewed or fat-tailed.

Actuarially Fair Price of Insurance Policy

A Finance 4335 student asked me the following question via email earlier today:

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):,

  1. on page 4 of the Supply of Insurance assigned 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”;
  2. on page 30 of the Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2) assigned 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
  3. on page 43 of the Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2) assigned reading, 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.”

More on the St. Petersburg Paradox…

Earlier this semester (specifically, on September 8) when we completed our coverage of the Decision Making under Risk and Uncertainty, part 1 lecture note, we learned that in 1738, the Swiss mathematician and physicist Daniel Bernoulli proposed his famous “St. Petersburg Paradox” in an academic paper entitled “Exposition of a New Theory on the Measurement of Risk“. As was the standard practice in academia at the time, Bernoulli’s article was originally published in Latin; fortunately, it was subsequently translated into English and published a second time 216 years later in Econometrica (1954, Volume 22, No. 1): pp. 22–36. This article made a great leap forward in the development of decision theory by introducing scholars to the expected utility model.

Also, the Wikipedia article about Bernoulli’s article is worth reading. It provides the mathematics for determining the price at which the apostle Paul would have been indifferent about taking the apostle Peter up on this bet. The original numerical example proposed by Bernoulli focuses attention on Paul’s gamble per se and does not explicitly consider the effect of Paul’s initial wealth on his willingness to pay. However, the quote on page 31 of the article (“… that any reasonable man would sell his chance … for twenty ducats”) implies that Bernoulli may have assumed Paul to be a millionaire, since (as shown in the Wikipedia article) the certainty-equivalent value of this bet to a millionaire who has logarithmic utility comes out to 20.88 ducats.

Midterm Exam 1 formula sheet and helpful hints

The formula sheet for Midterm Exam 1, which will be administered during class on Tuesday, September 27, is available at http://fin4335.garven.com/fall2022/formulas_part1.pdf.

Midterm Exam 1 consists of four problems. You will only be required to complete three of four problems. If you complete all four problems on the exam, only the three highest-scoring problems will count toward your Midterm Exam 1 grade.  Each problem will be worth 32 points, and you will receive 4 points for including your name on the exam booklet. Thus, the maximum number of points possible on Midterm Exam 1 will be 100.

Whenever you take an exam in Finance 4335, it is important to not only show your work but also provide complete answers for each question; i.e., besides producing appropriate numerical results, also clearly explain your results using plain English.

Here are some (which I think are) helpful hints for Midterm 1:

1. Risk Aversion, Risk Neutrality, and Risk-Loving Preferences.  Review definitions for risk aversion, risk neutrality, and risk-loving; see especially page 3 of http://fin4335.garven.com/fall2022/lecture6.pdf about these topics:

    • Risk-averse utility functions are characterized by diminishing marginal utility; thus, E(U(W)) < U(E(W));
    • Risk-neutral utility functions are characterized by constant marginal utility; thus, E(U(W)) = U(E(W)); and
    • Risk-loving utility functions are characterized by increasing marginal utility; thus, E(U(W)) > U(E(W)).

2. Risk Pooling.  I highly recommend reviewing the coverage of risk pooling on pp. 3-5 of the http://fin4335.garven.com/fall2022/lecture5.pdf lecture note, along with the Risk Pooling class problem.

3. Demand for insurance.  Under a “full coverage” policy, the insured pays a premium that transfers all risk to the insurer; if the premium charged for such coverage is actuarially fair, then the optimal choice for all arbitrarily risk-averse decision-makers is to purchase a full-coverage policy; this result is commonly referred to as the “Bernoulli Principle”.  See pp. 29-32 of the September 6th assigned reading entitled Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2).

4. Degree of risk aversion. For logarithmic and power utilities, we saw that all such utilities feature decreasing absolute risk aversion, which means that as one’s initial wealth increases, then one’s degree of aversion to a given risk declines; resulting in a lower risk premium at higher levels of initial wealth (see the discussion of Arrow-Pratt risk aversion measures on pp. 5-9 and 13-15 of http://fin4335.garven.com/fall2022/lecture6.pdf).

5. Stochastic Dominance.  If one risk stochastically dominates another, this implies that the expected utility for the dominating risk exceeds the expected utility for the dominated risk, a result that holds for all arbitrarily risk averse utilities (see the discussion of Stochastic Dominance on pp. 5-13 of http://fin4335.garven.com/fall2022/lecture8.pdf).

In my opinion, a careful review of the points I have made here, along with the information provided in the study guide entitled “Finance 4335 Midterm 1 Synopsis“, would be quite worthwhile as y’all prepare for the upcoming midterm exam.

Good luck!

Extra Credit Opportunity: Meet the Author Series, Featuring Dean Mazumder

On Tuesday, September 20, beginning at 3:30 pm in Foster 240, Dean Mazumder and his co-author Dale Cline will make a brief presentation of their book, “Money, Banking and Financial Markets: A Modern Introduction to Macroeconomics”.  You can earn extra credit by attending and reporting on their presentation and the Q&A session that follows.

If you decide to take advantage of this opportunity, I will use the grade you earn on your report to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest quiz grade). The report should be in the form of a 1-2 page executive summary in which you provide a critical analysis of this event. In order to receive credit, the report must be submitted via Canvas in either Word or PDF format by no later than Friday, September 23 at 5 p.m.

Extra credit opportunity for replacing lowest problem set grade

Class on Thursday, September 22, will be devoted to a Midterm Exam #1 Review Session.  In order to encourage everyone to prepare well for the review session, I have posted a Sample Midterm 1 Exam Booklet. If you complete this sample exam and turn it in on Canvas (see the link for this in the Assignments section) prior to the start of class that day, I will grade and count it as an extra credit problem set assignment which will replace your lowest problem set grade for Finance 4335 (assuming your grade on the sample midterm exam is higher).