# Problem Set 7 helpful hints

1. The least risky combination of Security A and Security B in Problem 1 is found by calculating ${w_A} = \displaystyle\frac{{\sigma _B^2 - {\sigma _{AB}}}}{{\sigma _A^2 + \sigma _B^2 - 2{\sigma _{AB}}}}$ and ${w_B} = 1 - {w_A}$.
2. It will always be the case for 2 security portfolios that by following the minimum variance portfolio weighting scheme in the previous bullet point, such a portfolio must have zero variance if ${\rho _{AB}} = 1$ or -1.
3. In part B of Problem 2, the Sharpe Ratio for security j is $\displaystyle\frac{{E({r_j}) - {r_f}}}{{{\sigma _j}}}$.

# Moral Hazard Extra Credit Problem due (via email sent to risk@garven.com) by 5 p.m. on Monday, October 16

I have written an extra credit problem set on the topic of moral hazard that will be due (via email sent to risk@garven.com) by 5 p.m. on Monday, October 16 (see http://fin4335.garven.com/fall2017/moralhazardEC.pdf). The grade received on this assignment will replace your lowest problem set grade, so long as it is higher than your lowest problem set grade.

Here are some helpful suggestions concerning how to go about working this problem set. The key learning objective for this problem set is to understand how compensation contract design affects managerial incentives to look after the interests of the firm’s owners. For simplicity, I assume here that investors are risk neutral; thus, they expect the manager to maximize the expected value of profit. Put yourself in the shoes of a compensation consultant and show the relationship between contract designs and expected profits under the various compensation scenarios that are provided in the problem set.

At your option, you may solve this problem set by hand or by creating a spreadsheet model. If you rely upon a spreadsheet model, you’ll need to include the spreadsheet as a file attachment when you send it to the risk@garven.com email address. Although a spreadsheet model is not required, you’ll find that Solver will come in very handy, particularly for answering the questions posed in parts 3-5 of the problem set.

# Problem Set 6 Hints and Spreadsheet

In question 1, part C of Problem Set 6, I ask you to “Find the maximum price which the insurer can charge for the coinsurance contract such that profit can still be earned while at the same time providing the typical Florida homeowner with higher expected utility from insuring and retrofitting. How much profit will the insurer earn on a per policy basis?” Here are some hints which you’ll hopefully find helpful.

For starters, keep in mind that in part A, insurance is compulsory (i.e., required by law), whereas, in parts B and C, insurance is not compulsory. In part B of question 1, you are asked to show that the homeowner retrofits while choosing not to purchase insurance. However, in part C, the homeowner might retrofit and purchase coverage if the coinsurance contract is priced more affordably. Therefore, the insurer’s problem is to figure out how much it must reduce the price of the coinsurance contract so that the typical Florida homeowner will have higher expected utility from insuring and retrofitting compared with the part B’s alternative of retrofitting only. It turns out that this is a fairly easy problem to solve using Solver. If you download the spreadsheet located at http://fin4335.garven.com/fall2017/ps6.xls, you can find the price at which the consumer would be indifferent between buying coinsurance and retrofitting compared with self-insurance and retrofitting. You can determine this “breakeven” price by having Solver set the D28 target cell (which is expected utility for coinsurance with retrofitting) equal to a value of 703.8730 (which is the expected utility of self-insurance and retrofitting) by changing cell D12, which is the premium charged for the coinsurance. Once you have the “breakeven” price, all you have to do to get the homeowner to buy the coinsurance contract is cut its price slightly below that level so that the alternative of coinsurance plus retrofitting is greater than the expected utility of retrofitting only. The insurer’s profit then is simply the difference between the price that motivates the purchase of insurance less the expected claims cost to the insurer (which is $1,250). # Problem set 5 questions from a Finance 4335 student… A student asked me the following questions about Problem Set 5 (due at the beginning of class on Tuesday, October 3): Question 1: “I am having trouble with Problem Set 5. What exactly does Part A mean when it asks for premium loading? I cannot seem to recall in my notes what exactly that is and how it applies to this problem.” My Answer to Question 1: In insurance, the premium loading corresponds to the “markup” from the actuarially fair value. Part A asks for the premium loading in dollar and percentage terms, so you need to figure out what the actuarially fair value is for the policy and compare that to the quoted price. Question 2: “And for Part B, am I right to assume that the “optimal” level of insurance coverage is being calculated with the$240 insurance premium that is given in the problem?”

My Answer to Question 2: Yes.

# Office Hours & Problem Sets

Hello Class,

I should have your final problem sets graded by Friday, so if you wish to pick them up to help with your studies, I plan on hosting office hours between 12:00 to 2:00 pm on Friday at the Financial Markets Center. I may also be available after 5:00 that day, if you wish to ask questions or pick up your problem sets then you should email me beforehand. Provided you study problem sets 3 & 4 as well as the practice midterm, you should be well prepared for the exam, which is structured in the same manner.

However, if you do not understand how to apply the concepts learned in class, nor how to utilize derivatives or standard normal distributions, this and other exams will prove quite difficult. I recommend at a minimum using a graphing calculator such as  TI-84, TI-89, or TI-Nspire CAS, but it is not necessary, albeit very helpful. Furthermore, DO NOT round any intermediate calculations, otherwise you may get the wrong answer.

There have been times when the precision level of the final answer results in very difficult outcomes if the intermediate calculations are rounded. Additionally, show all relevant work, or you may not receive partial credit. Answers that are close but mechanically wrong are still incorrect, but answers that are close due to rounding error may receive credit depending on the severity. Make sure you always answer the question asked, not tangential information.

I believe that each of you are capable of performing well in the class, as long as you do the appropriate practice and check your work carefully. Many of you have performed exceptionally on the problem sets so far, but some of you likely will need to dedicate significant time to really understand the material. Furthermore, even if you did well on the problem sets, you should still study them to ensure you have not forgot the material. Good luck!

-Alexander Law

# How to best prepare for the midterm exam review session on Thursday…

Class on Thursday, September 21 will be devoted to 1) completing the “Decision-Making under Risk and Uncertainty” topic and 2) a review session for Midterm Exam 1, which is scheduled for Tuesday, September 26 (in class). Prior to coming to class on Thursday, you’ll want to review your notes about the “Decision-Making under Risk and Uncertainty” topic, review problem sets 3 and 4, and try solving the Finance 4335 Midterm 1 Exam that I gave during the Spring 2017 semester (solutions are available at http://fin4335.garven.com/fall2017/midterm1_spring2017_solutions.pdf). This way we can have a very productive review session that will enable you to be better prepared for the first midterm in Finance 4335!

# Hints for solving problem set #4 (Hint #2)

A student asked me whether it is possible for the value of α in problem set 4 to be negative. Here, α represents the optimal level of exposure to the risky asset; 1-α represents the optimal level of exposure to the safe bond. While it is certainly theoretically possible for α to have a negative value, for this particular problem it turns out that α > 0. The reason α is positive in this case is because the 60/40 probabilities imply a positive expected return on the risky asset which exceeds the expected return on the bond. Thus, even though the stock is risky, a square root utility investor is willing to invest some of her money in the stock because in an expected utility sense, some positive exposure to risk is worthwhile.

If you obtain a negative value for α, this means that you must have made a math error somewhere. Since E(U(W)) = .6(105 + 25α ).5 + .4(105 – 35α ).5, then one can find the optimal value for a by applying the chain rule individually to both the .6(105 + 25α ).5 and the .4(105 – 35α ).5 terms, setting the resulting equation equal to zero (this is the so-called “first order condition”) and solving for α .

If you get stuck on the math at all, you might consider inputting the data into an Excel spreadsheet and use Solver to find the optimal value for α . For what it’s worth, I just did this a few moments ago and confirmed that the (positive) α value which I obtained using the calculus is identical to the α value indicated by Excel (obviously I was not surprised, since in both cases I knew a priori that my calculus and Excel code were correct :-)).

This raises an interesting question; specifically, what would have to be different about this problem in order to obtain a negative value for α ? If this problem were reparameterized such that the risk/return characteristics of the stock vis-a-vis the bond were sufficiently worsened, and/or if the investor was inclined to act in a more risk averse fashion (e.g., if initial wealth declined and/or the investor’s utility function were different), then a negative value for α is a possibility. For example, suppose that everything stayed the same, but that the state probabilities for the stock were 55/45 rather than 60/40. If this occurred, then you would find that the investor’s optimal α is -48.33%. Note that with 55/45 state probabilities, the stock has an expected return of .55(.3) + .45 (-.3) = 3% and a standard deviation of 14.92% (check this calculation for yourself). If short selling were prohibited, then this investor would optimally invest all of her initial wealth in the bond. However, if short selling were allowed, then at date 0, she would sell short $48.33 of stock and invest her initial wealth of$100 plus the $48.33 in proceeds from the short sale in the bond. From date 0 to date 1, she would earn 5%, or$7.42 on her $148.33 bond investment. At date 1, she would close out her short position by buying the stock back at either$48.33 x (1.30) = $62.83 (in which case she would lose$7.08 on her $100 net investment) or at$48.33 x (.7) = $33.83 (in which case she would gain$21.92 on her $100 net investment). Thus the expected return on her portfolio is .55(-7.08%) + .45(21.92%) = 5.97%, and the standard deviation is 14.43% (short selling is risky because you might get stuck having to close out the short position at a high price; this is why the standard deviation is so high). I have posted my spreadsheet for this problem at http://risk.garven.com/wp-content/uploads/2017/09/assetallocation.xls. You are welcome to use this spreadsheet if you are interested in numerically validating your calculus-based solution procedure. # Hints for solving problem set #4 (Hint #1) Problem set #4 involves determining how to (optimally) allocate your initial wealth W0 =$100 to (risky) stock and (safe) bond investments. Let α represent the allocation to stock; then the plan is to invest $100α in the stock and$100(1-α) in the bond. The key here is to find the value for a which maximizes expected utility. The problem is based on the following facts:

• U(W) = W.5;
• W0 = \$100;
• current bond and stock prices are B0 and S0 respectively;
• end-of-period bond price is B1 = B0(1.05) with probability 1.0; and
• end-of-period stock price is S1 = S0(1.3) with probability .6 and S1 = S0(.7) with probability .4.

In order to compute expected utility of wealth, you must first determine state-contingent wealth (Ws). Since there is a 60% chance that the stock increases in value by 30%, a 40% chance that the stock decreases 30%, and a 100% chance that the bond increases in value by 5%, this implies the following:

• 60% of the time, Ws = αW0(1.3) + (1-α)W0(1.05) = α100(1.30) + (1-α)100(1.05) = α130 + (1-α)105 = 105 + 25α.
• 40% of the time, Ws = αW0(.7) + (1-α)W0(1.05) = α100(.7) + (1-α)100(1.05)] = α70 + (1-α)105 = 105 – 35α.

Therefore, expected utility is: E(U(W)) = .6(105 + 25α).5 + .4(105 – 35α).5. It is up to you to solve for the optimal value of α. This requires solving the first order condition, which involves differentiating E(U(W)) with respect to α, setting the result equal to 0 and solving for α.