# Tomorrow’s review session and various other study hints for the final exam in Finance 4335

I plan to devote tomorrow’s Finance 4335 class meeting to a review session for the final exam. If time permits, be sure to look over the Final Review and Summary of Finance 4335 lecture note prior to the start of class tomorrow.  This document summarizes virtually all the topics we covered this semester in Finance 4335. I designed this document to serve as a study guide to help you prepare for the final exam.

Here are some other helpful study hints to consider:

# Credit risk Q&A with a Finance 4335 student

Yesterday afternoon, I received the following email from a Finance 4335 student about credit risk:

Date: Friday, November 20, 2020 at 3:40 PM
To: James Garven <James_Garven@baylor.edu>
Subject: Question for Problem Set 9

Doctor Garven,

I just have a question for part 3 of the problem set 9 which is due next Tuesday. I’m confused as to why the fair premium of the insurance corresponds to dollar value of the limited liability put option. And if the firm purchases the insurance, will its credit risk premium decline to 0?

Perhaps I should more clearly define “fair” premium. In this setting the “fair” insurance premium corresponds to the premium paid by the firm such that it is indifferent between buying and not buying credit insurance.

Consider the following numerical example. Suppose the riskless rate of interest is zero, and a firm promises to pay $100 one year from now. Then the yield to maturity of the debt is also zero, and the present value is$100. Next, suppose an otherwise identical and uninsured firm issues debt with a promised repayment of $100, but it has a high probability of default so the current market value of the debt is only$80; in this latter case, the yield to maturity and the credit risk premium are the same, since YTM = r + credit risk premium and r = 0. Thus, YTM = credit risk premium = 25% (here I am assuming annual compounding for the sake of simplicity, so FV = PV(1+YTM) ==>100 = 80(1.25)). Now, suppose this uninsured firm changes its mind and purchases full insurance coverage for the “fair” premium of $20 (which corresponds to the value of the limited liability put option). Since the insured firm’s debt is no longer risky to investors, investors currently value the firm’s debt issue at its par value of$100, which implies that the credit risk premium falls to 0. However, from the standpoint of the firm, it can only expect to net $80 from its bond issue, irrespective of whether it purchases credit insurance, so it is indifferent between buying and not buying credit insurance. In the real world, credit risk enhancement is a viable financial services business because so-called “prudent-person” rules often severely limit the marketability of sub-investment grade credit to institutional investors. Thus, credit enhancement to investment grade can add more value for firms, NGO’s, and government organizations by substantially expanding the market for potential investors in such credit issues. # Formula Sheet for Midterm 2 (along with some Midterm 2 – related information) The formula sheet for Midterm Exam 2 is now available at http://fin4335.garven.com/fall2020/formulas_part2.pdf. This document will also be linked to Canvas when you take the exam on Tuesday. In my opinion, the two most important Midterm 2 – related documents are the formula sheet and the Finance 4335 course synopsis for Midterm Exam 2. Not surprisingly, the topics on the exam include insurance economics, asymmetric information, portfolio theory, and capital market theory. # Important Midterm 1 reminders • My first reminder is that all previous Midterm 1 – related blog postings are linked @ http://risk.garven.com/?s=Midterm+1. • I require all students to be logged in to the regularly scheduled class meeting on Zoom with their video cameras turned on (unmuted) throughout the exam. I will start our class Zoom sessions tomorrow no later than 10:50 am for Section 1 and no later than 1:50 pm for Section 2 so that students can ask last-minute questions prior to the start of the exam. • I have posted Midterm 1 on Canvas; it is located on the Quizzes page under the name “FIN 4335 – Midterm Exam #1”. You will have a total of 85 minutes to take the exam and also post a single PDF file comprising your written work in support of the answers that you provide in Canvas. The window for taking the exam is 95 minutes; from 10:55 am CT until 12:30 pm CT for Section 1, and from 1:55 pm CT until 3:30 pm CT for Section 2. Once you have opened your exam up, then your session will automatically time out after 85 minutes (I assume that all of you will begin by no later than 11:05 am in Section 1 or 2:05 pm in Section 2). • Plan on allocating 75 minutes to complete the exam, and 10 minutes to creating and uploading your PDF file. • The first “Question” on this exam is an “Honor Pledge”, which requires that you certify that all work on this exam is your own. If you do not complete the honor pledge, then you will receive a zero on this exam. • This exam comprises 3 problems worth 32 points each; these problems appear as “Questions” 2-4. • The final “Question”, labeled “Question 5”, is where you upload your written work. Do not, under any circumstance, click on the ”Submit Quiz” button prior to completing the exam and uploading your PDF file. If you submit your completed exam without your PDF file, then you will receive a grade of zero for this exam. • Since the number of points adds up to 96, students who complete this exam will also receive an additional 4 points. Thus, the maximum number of points possible for this exam is 100. 🙂 • If you have any questions or concerns during this exam, call or text Dr. Garven at 254-307-1317. # Midterm 1 formula sheet, helpful hints, pre-exam office hours I just posted the formula sheet for the Midterm 1 exam. Actually, the formula sheet consists of two pages; the first page is a formula sheet, and the second page is a standard normal distribution table. The exam consists of 3 problems worth 32 points each; I add 4 points to everyone’s scores so that that the maximum number of points possible is 100. On Tuesday, plan on allocating no more than 75 minutes to complete the exam, and no more than 10 minutes to upload a single PDF of your written work which clearly demonstrates your conceptual grasp and ability to clearly explain, in plain English, how you arrived at all of your answers on the exam. You may use either a calculator or a spreadsheet for any computations that are required for the exam. Here are some (what I think are) helpful hints: 1. Review definitions for risk aversion, risk neutrality, and risk-loving behavior; see especially page 2 of http://fin4335.garven.com/fall2020/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. Full coverage insurance: Under a “full coverage” policy, the insured pays a premium which 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 “Bernoulli Principle” (see http://risk.garven.com/2020/09/16/fair-price-of-insurance-policy/). 3. 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, the one’s degree of aversion to a given risk declines; resulting in a lower risk premium at higher levels of initial wealth (see pp. 9-12 of http://fin4335.garven.com/fall2020/lecture6.pdf). Tomorrow, I will be available in my virtual Zoom office from 3-5 pm CT in case if any students would like to stop by for a pre-exam chat. Good luck! # Preparing for the first midterm in Finance 4335: Review Session (9/24) and Exam (9/29) On September 24 and September 29, we’ll focus our attention on the first midterm exam in Finance 4335. Class on Thursday, September 24 will be devoted to a review session for the exam, and the exam will be administered during class on Tuesday, September 29. Even though I list it as “optional” reading, I highly recommend that y’all read my teaching note due on 9/24 entitled “Finance 4335 course synopsis for Midterm Exam 1”. In order to properly prepare for the first midterm exam in Finance 4335, I also recommend reviewing the various class problems along with problem sets 2 – 4, and midterm 1 from Spring 2020; solutions for class problems and problem sets in FIN 4335 are available at http://risk.garven.com/category/problem-set-solutions. After class tomorrow, I will post the solutions for problem set 4 and for midterm 1 from Spring 2020 so that everyone will the opportunity to check their work on these assignments. # Hints for solving problem set #4 (spreadsheet hint) An acceptable alternative way (either in lieu of or in addition to) for solving problem 2 in problem set #4 would be to build a spreadsheet model in which you use Solver to determine the optimal exposure to risk for both investors. If you decide to build your own spreadsheet model, upload it in addition to the problem set itself in order to get credit for working problem 2 this way. An example of how to use Solver for figuring out an optimal decision is provided in my “Optimal insurance demand” spreadsheet. There, alpha corresponds to the percent of the risk to be insured and beta corresponds to the percentage markup from the actuarially fair price; when beta is zero (as currently coded), then insurance is actuarially fair and Solver returns an alpha value of 1 (i.e., full coverage; as currently coded, this spreadsheet corroborates the Bernoulli principle for a consumer with a square root utility function. # Hints for solving problem set #4 (calculus hint) In a previous “helpful” hint pertaining to the second problem in Problem Set 4, I noted (among other things) that expected utility for Investor A is E(U(W)) = .6(1,020 + 100x).5 + .4(1,020 – 140x).5, where x corresponds to the proportion of the portfolio that is to be allocated to the risky asset. “Optimal” exposure to risk is determined by maximizing E(U(W)); this is accomplished by differentiating E(U(W)) with respect to x, setting that result equal to 0, and solving for x. The mathematical logic applied here is the same as the approach shown on pp. 14-15 in the Mathematics Tutorial for determining the profit-maximizing production decision of a firm. Here, since the square root utility function is itself a function of x; i.e., U(W(x)), this means that we must apply the chain rule in order to differentiate E(U(W)) with respect to x: $\displaystyle\frac{{dE(U(W))}}{{dx}} = \frac{{dE(U(W))}}{{dW}}\frac{{dW}}{{dx}} = .5W{(x)^{ - .5}}\frac{{dW}}{{dx}}.$ Since there are two states, this means that the are two state-contingent values for W(x); specifically, 60% of the time, W(x) = 1,020 + 100x , and 40% of the time, W(x) = 1,020 -140x. Once the chain rule has been applied to differentiating both terms on the right-hand side of the E(U(W)) equation for Investor A, set that result (also known as the “first-order condition”) equal to 0 and solve for Investor A’s optimal exposure to the risky asset. Once that proportion has been determined, the allocation to the safe asset is equal to 1-x. Rinse and repeat to determine Investor B’s optimal exposure to risk. Since Investor B’s utility function is E(U(W)) = .6ln (1,020 + 100x) + .4ln(1,020 – 140x), it follows that the first-order condition is $\displaystyle \frac{{dE(U(W))}}{{dx}} = \frac{{dE(U(W))}}{{dW}}\frac{{dW}}{{dx}} = \frac{1}{{W(x)}}\frac{{dW}}{{dx}}.$ Set Investor B’s first-order condition equal to 0, solve for x and 1-x, and you’re good to go! # Hints for solving problem set #4 Problem set #4 (which is due on Thursday, September 24) consists of two problems: 1) a problem that incorporates both expected utility and stochastic dominance, and 2) an optimal (expected utility-maximizing) portfolio problem. We’ll discuss stochastic dominance in class next Tuesday, but in the meantime, here are some hints for setting up the second problem. The second problem involves determining how to (optimally) allocate initial wealth W0 =$1,000 to (risky) stock and (safe) bond investments for two investors who are identical in all respects except utility. Let x represent the allocation to stock; then the plan is to invest $1,000x in the stock and$1,000(1-x) in the bond. The key here is to find the value for x which maximizes expected utility. The problem is based on the following facts:

• U(W) = W.5; for Investor A and U(W) = ln W for Investor B;
• W0 = \$1,000 for both investors;
• Current bond and stock prices are B0 and S0 respectively;
• End-of-period bond price is B1 = B0(1.02) with probability 1.0; and
• End-of-period stock price is S1 = S0(1.12) with probability .6 and S1 = S0(.88) with probability .4.

In order to compute expected utility of wealth for either investor, you must first determine state-contingent wealth (Ws). Since there is a 60% chance that the stock increases in value by 12%, a 40% chance that the stock decreases in value by 12%, and a 100% chance that the bond increases in value by 2%, this implies the following:

• 60% of the time, Ws = xW0(1.12) + (1-x)W0(1.02) = x1,000(1.12) + (1-x)1,000(1.02) = x1,120 + (1-x)1,020 = 1,020 + 100x.
• 40% of the time, Ws = xW0(.88) + (1-x)W0(1.02) = x1,000(.88) + (1-x)1,000(1.02) = x880 + (1-x)1,020 = 1,020 – 140x.

Therefore, expected utility for Investor A is: E(U(W)) = .6(1,020 + 100x).5 + .4(1,020 – 140x).5, and expected utility for Investor B is E(U(W)) = .6ln (1,020 + 100x) + .4ln(1,020 – 140x). There are two ways to solve for the optimal value of x for each investor – via calculus or a spreadsheet model; either approach suffices.

# Fair Price of Insurance Policy

A Finance 4335 student asked the following question earlier today:

Q: “I have a quick homework question. How do you find the fair price of an insurance policy?”

Here’s the answer I provided, which I now share with all Finance 4335 students:

A: “I believe you are referring to the “actuarially fair” price, or insurance premium. The actuarially fair 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, are explained in two of the September 8th assigned readings; e.g.,

1. 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”;
2. 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
3. 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.””