# Some observations concerning the Rothschild-Stiglitz numerical example from today’s class…

Today, we considered the following problem in class:

• Assume that consumers are identical in all respects expect for their loss probabilities; some are high risk, and others are low risk.
• Members of the high-risk group have loss probability pH = 65%, whereas members of the low risk group have loss probability pL = 35%.
• Each consumer has initial wealth of $100 and utility U(W)=W.5. • There are only two possible states of the world, loss and no loss. If a loss occurs, then consumers lose their initial wealth of$100.
• Insurance contract offerings include the following:
• Policy A provides full coverage for a price of $65. • Policy B provides full coverage for a price of$45.50.
• Policy C provides 60% coverage for a price of $39. • Policy D provides 30% coverage for a price of$13.65.

# Analytic and Numerical Proofs of the Bernoulli Principle and Mossin’s Theorem

During yesterday’s class meeting, we discussed two particularly important insurance economics concepts: 1) the Bernoulli Principle, and 2) Mossin’s Theorem. The Bernoulli Principle states that if an actuarially fair full coverage insurance policy is offered, then an arbitrarily risk averse individual will purchase it, and Mossin’s Theorem states that if insurance is actuarially unfair, then an arbitrarily risk averse individual will prefer partial insurance coverage over full insurance coverage.

The Bernoulli Principle is graphically illustrated in the following figure (taken from p. 4 in the Insurance Economics lecture note):

Here, the consumer has initial wealth of $120 and there is a 25% chance that a fire will occur which (in the absence of insurance) will reduce her wealth from$120 to $20. The expected loss is E(L) = .25(100) =$25, which is the actuarially fair price for a full coverage insurance policy. Thus if she remained uninsured, she would have E(W) = .25(20) + .75(120) = $95, and her utility would be EU. Note that the (95, EU) wealth/utility pair corresponds to point C in the figure above. Now suppose the consumer purchases an actuarially fair full coverage insurance policy. Since she is fully insured, she is no longer exposed to any risk; her net worth is$95 irrespective of whether the loss occurs, her utility is U(95), and her wealth/utility pair is located at point D in the figure above. Since U(95) > EU, she will fully insure. More generally, since this decision is optimal for an arbitrarily risk averse consumer, it is also optimal for all risk averse consumers, irrespective of the degree to which they are risk averse.

Next, we introduced coinsurance. A coinsurance contract calls for proportional risk sharing between the consumer and the insurer. The consumer selects a coinsurance rate $\alpha$, where $\alpha$ represents the proportion of loss covered by the insurer. By definition, $\alpha$ is bounded from below at 0 and from above at 1. Thus, if the consumer selects $\alpha$ = 0, this means that she does not purchase any insurance (i.e., she self-insures). If she selects $\alpha$ = 1, this implies that she obtains full coverage. Furthermore, the price of a coinsurance contract is equal to the price of a full coverage insurance contract (P) multiplied by $\alpha$. On pp. 6-7 of the Insurance Economics lecture note, I analytically (via the calculus) confirm the Bernoulli principle by showing that if the consumer’s utility is U = W.5 and insurance is actuarially fair, then the value for $\alpha$ which maximizes expected utility is $\alpha$ = 1. An Excel spreadsheet called the “Bernoulli and Mossin Spreadsheet” is available from the course website which enables students to work this same problem using Solver. I recommend that you download this spreadsheet and use Solver in order to validate the results for a premium loading percentage (β) equal to 0 (which implies a full coverage (actuarially fair) premium P = E(L)(1+β) =$25 x (1+0) =$25) and β = 0.6 (which implies an actuarially unfair premium of $40, the analytic solution for which is presented on p. 8 in the Insurance Economics lecture note). Let’s use Solver to determine what the optimal coinsurance rate ($\alpha$) is for U = W.5 when β = 0 (i.e., when insurance is actuarially fair and full coverage can be purchased for the actuarially fair premium of$25). To do this, open up the Bernoulli and Mossin Spreadsheet and invoke Solver by selecting “Data – Solver”; here’s how your screen will look at this point:

This spreadsheet is based upon the so-called “power utility” function U = Wn, where is 0 < n < 1. Since we are interested in determining the optimal coinsurance rate for a consumer with U = W.5, we set cell B2 (labeled “exponent value”) equal to 0.500. With no insurance coverage (i.e., when $\alpha$ = 0) , we find that E(W) = $95 and E(U(W)) = 9.334. Furthermore, the standard deviation (σ) of wealth is$43.30. This makes sense since the only source of risk in the model is the risk related to the potentially insurable loss.

Since we are interested in finding the value for $\alpha$ which maximizes E(U(W)), Solver’s “Set Objective” field must be set to cell B1 (which is a cell in which the calculated value for E(U(W)) gets stored) and Solver’s “By Changing Variable Cells” field must be set to cell B3 (which is the cell in which the coinsurance rate $\alpha$ gets stored). Since insurance is actuarially fair (β = 0 in cell B4), the Bernoulli Principle implies that the optimal value for $\alpha$ is 1.0. You can confirm this by clicking on Solver’s “Solve” button:

Not only is $\alpha$ = 1.0, but we also find that E(W) = $95, σ = 0 and E(U(W)) = 9.75. Utility is higher because this risk averse individual receives the same expected value of wealth as before ($95) without having to bear any risk (since σ = 0 when $\alpha$ = 1.0).

Next, let’s determine what the optimal coinsurance rate ($\alpha$) is for U = W.5 when β = 0.60 (i.e., when insurance is actuarially unfair). As noted earlier, this implies that the insurance premium for a full coverage policy is $25(1.60) =$40. Furthermore, the insurance premium for partial coverage is $$\alpha$40. Reset $\alpha$ ’s value in cell B3 back to 0, β’s value in cell B4 equal to 0.60, and invoke Solver once again: On p. 8 in the Insurance Economics lecture note, we showed analytically that the optimal coinsurance rate is 1/7, and this value for $\alpha$ is indicated by clicking on the “Solve” button: Since $\alpha$ = 1/7, this implies that the insurance premium is (1/7)40 =$5.71 and the uninsured loss in the Fire state is (6/7)(100) = $85.71. Thus in the Fire state, state contingent wealth is equal to initial wealth of$120 minus $5.71 for the insurance premium minus for$85.71 for the uninsured loss, or $28.57, and in the No Fire state, state contingent wealth is equal to initial wealth of$120 minus $5.71 for the insurance premium, or$114.29. Since insurance has become very expensive, this diminishes the benefit of insurance in a utility sense, so in this case only a very limited amount of coverage is demanded. Note that E(U(W)) = 9.354 compared with E(U(W)) = 9.334 if no insurance is purchased (as an exercise, try increasing β to 100%; you’ll find in that case that no insurance is demanded (i.e., $\alpha$ = 0).

I highly recommend that students conduct sensitivity analysis by making the consumer poorer or richer (by reducing or increasing cell B5 from its initial value of 120) and more or less risk averse (by lowering or increasing cell B2 from its initial value of 0.500). Other obvious candidates for sensitivity analysis include changing the probability of Fire (note: I have coded the spreadsheet so that any changes in the probability of Fire are also automatically reflected by corresponding changes in the probability of No Fire) as well as experimenting with changes in loss severity (by changing cell C8).

# 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. # Arrow-Pratt method vis–à–vis the “exact” method for calculating risk premiums I received an email from a Finance 4335 student earlier today asking for some clarification regarding the Arrow-Pratt method vis–à–vis (what I like to refer to as) the “exact” method for calculating risk premiums. As I showed in class, the Arrow-Pratt method is an alternative method for calculating the risk premium. Thus, either approach (the “exact” method or the Arrow-Pratt method) is perfectly acceptable for calculating risk premiums. The value added of Arrow-Pratt is (as I point out in my two page Finance 4335 synopsis) that it analytically demonstrates how risk premiums depend upon two factors: 1) the magnitude of the risk itself (as indicated by variance), and 2) the degree to which the decision-maker is risk averse. For example, the Arrow-Pratt coefficient for the logarithmic investor (for whom U(W) = ln W) is twice as large as the Arrow-Pratt coefficient for the square root investor (for whom U(W) = W.5); 1/W for the logarithmic investor compared with .5/W for the square root investor. Thus, the logarithmic investor behaves in a more risk averse than the square root investor; other things equal, the logarithmic investor will prefer to allocate less of her wealth to risky assets and buy more insurance than the the square root investor. Another important insight yielded by Arrow-Pratt (at least for the types of utility functions we have considered in Finance 4335; i.e., power and logarithmic utilities) is the notion of decreasing absolute risk aversion. Other things equal, investors become less (more) risk averse as wealth increases (decreases). # Finance 4335 Midterm Exam 1 information I have just finished writing Midterm Exam 1 for Finance 4335. This test consists of 2 problems worth 32 points each and 1 problem worth 36 points. Thus, the maximum number of points possible for this exam is 100. This exam will be given during our scheduled class time on Tuesday, September 26, from 11 a.m. – 12:15 p.m. in Foster 402. I also just uploaded the formula sheet that will be included as part of the exam booklet on Tuesday; I highly recommend that you download and review the “Formula Sheet for Midterm exam #1” (technical note: if you have previously clicked on this link, clear out your browser cache so as to ensure that you are able to access the current version of this document). Not surprisingly, the exam is all about the “Decision Making under Risk and Uncertainty” topic. In my opinion, the best way to prepare for the exam is to review my updated two page synopsis of what we have covered in Finance 4335 since the start of this semester. Also, review the third and fourth problem sets, as well as the Sample Midterm 1 Exam Booklet and solutions. Anyway, best of luck on the exam on Tuesday. If you have any questions or concerns, don’t hesitate to call me at my Baylor office number, which is 254-710-6207. # 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.