As previously announced, I plan to hold office hours (in Foster 320.39) from 10 a.m. – 12 noon. If you have any questions or concerns about Finance 4335, that would be a good time to stop by. Even if those hours are not convenient for you, feel free to call me during business hours on any day of the week. Calls placed to my Baylor faculty office phone (254-710-6207) are automatically routed to my cellphone as well as to my home office, so I am pretty easy to contact even when I am not on campus.
To find the “optimal” coinsurance rate for either utility function referenced in Problem Set 5, the first step is to determine state-contingent wealth for the loss and no loss states. Let be the coinsurance rate (note that cannot be less than 0 or greater than 1; implies that you self-insure, whereas implies that you purchase full coverage). Since there is a 20% chance of a $1,000 loss and a 80% chance that there is no loss, this implies that state-contingent wealth ( ) is calculated as follows:
- 20% of the time, , where P is the price of full insurance coverage = $240. Therefore, .
- 80% of the time, .
Therefore, the expected utility equation that you need to maximize (by selecting the appropriate coinsurance rate) is:
All that you need to do is to apply the two utility functions provided in this problem (specifically, and ) and then differentiate each expected utility equation with respect to , set the result equal to 0, and determine the optimal via algebra.
I also highly recommend that you try your hand at recoding the Bernoulli and Mossin Spreadsheet to accommodate the parameters given in Problem Set 5. This way, you can validate the results that you obtain by applying the calculus.
Finally, a “shortcut” for answering part D of Problem Set 5 involves calculating the Arrow-Pratt absolute risk aversion coefficient for each utility function. That will indicate which utility is more risk averse, and logically the more risk averse utility will, for a given premium leading optimally select a higher .
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.
We graphically illustrated the Bernoulli Principle by using 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 α, where α represents the proportion of loss covered by the insurer. By definition, α is bounded from below at 0 and from above at 1. Thus, if the consumer selects α = 0, this means that she does not purchase any insurance (i.e., she self-insures). If she selects α = 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 α. 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 α which maximizes expected utility is α = 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 (α) 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 α = 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 α 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 α gets stored). Since insurance is actuarially fair (β = 0 in cell B4), the Bernoulli Principle implies that the optimal value for α is 1.0. You can confirm this by clicking on Solver’s “Solve” button:
Not only is α = 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 α = 1.0).
Next, let’s determine what the optimal coinsurance rate (α) 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 $α40. Reset α ’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 α is indicated by clicking on the “Solve” button:
Since α = 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., α = 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).
Here are the Finance 4335 reading and viewing assignments for Friday, February 24th. Class will begin that day with a quiz based upon these assignments:
1. Moral Hazard & Adverse Selection (Doherty, Chapter 3)
2. Moral Hazard & Adverse Selection Synopsis, by James R. Garven
3. View the following five YouTube videos: a) Asymmetric Information and Used Cars, b) Asymmetric Information and Health Insurance, c) Moral Hazard, d) Solutions to Moral Hazard, e) Signaling
Here’s a result that we discussed in class regarding power utilities that I think you’ll find helpful when you are working on the midterm exam tomorrow morning. Specifically, the square root utility that we frequently used is an example of a risk averse “power” utility. It is risk averse because marginal utility () is positive and diminishing ( < 0). The Arrow-Pratt coefficient is the negative of the ratio of the 2nd over the 1st derivative, which means that for the square root function, the Arrow-Pratt coefficient is equal to .5/W.
The more general case of the risk averse power utility is , where n is a number between 0 and 1. This utility is risk averse because marginal utility () is positive and diminishing ( < 0). Furthermore, the Arrow-Pratt coefficient for this general case is equal to (1-n)/W. Thus, if n is equal to say, .9, then the the Arrow-Pratt coefficient for this particular case is equal to .1/W; this utility is considerably less risk averse than the square root utility. On the other hand, if n = .1, then the Arrow-Pratt coefficient for such at utility is equal to .9/W; this utility is considerably more risk averse than the square root utility.
Other cases of possible interest include risk neutrality, where n = 1. In that case, the Arrow-Pratt coefficient is equal to 0, which implies that the risk premium is also equal to 0. If n > 1, then this implies a negative Arrow-Pratt coefficient, which in turn implies that the investor is risk loving because the risk premium is also negative. Another risk averse utility that we have studies is the logarithmic utility, which features an Arrow-Pratt coefficient is equal to 1/W; thus, all risk averse power utilities are less risk averse than the logarithmic utility.
I think it should go without saying (but I’ll say it anyway J) that I welcome calls at my Baylor office number (254-710-6207) from any of you who may have questions or concerns about tomorrow’s midterm specifically and Finance 4335 generally… If not, I’ll look forward to seeing all of you bright and early tomorrow morning at 9:05 a..m. in Foster 203!
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).
Here is some information about the midterm exam that will be given tomorrow morning during the first part of the Finance 4335 class meeting (the second part of the class meeting will be devoted to covering the Insurance Economics topic, the assigned reading for which is The Demand for Insurance).
The exam consists of a total of three problems worth 32 points each, plus 4 points for including your name on the exam booklet for a total of 100 points. Not surprisingly, the exam is all about the “Decision Making under Risk and Uncertainty” topic that we have covered during the course of the last three meetings of Finance 4335. I have posted a copy of the formula sheet that will appear as part of the exam booklet at http://fin4335.garven.com/spring2017/formulas_part1.pdf; it would be a good idea to familiarize yourself with this document prior to tomorrow’s exam.
In my opinion, the best way to prepare for the exam is to review my two page synopsis of what we have covered in Finance 4335 to date. Also, review the third and fourth problem sets, as well as the Sample Midterm 1 Exam Booklet and solutions.
Problem 3 is illustrated in the worksheet labeled “Optimal Portfolio.” By simply adjusting the “Percent of Portfolio” proportions, you can generate all of the answers for the various crop mixes suggested in problem 3. You can also determine the expected utility maximizing portfolio using Solver (which turns out to be 71.87% Potatoes, 28.13% Strawberries).
Problem 4 is illustrated in the worksheet labeled “Venture Capital.” Three separate tables are presented there: 1) “Gary on his own,” 2) “Jill’s utility from partnership,” and 3) “Gary’s utility from partnership.” As we saw in class, Gary prefers to be a sole proprietor if Jill can acquire her 50% share of the business for a price of $10,000. This is indicated by the following screenshot, which shows that Gary’s EU from a sole proprietorship is higher than his EU from an equal partnership based upon a price of $10,000:
However, Gary might prefer an equal (50/50) partnership if he can fetch a higher price for selling half of the business to Jill. In order to determine the price at which Jill is indifferent between investing in the partnership and not investing can be determined by finding the price at which participating in the partnership provides the same expected utility as not participating (which is 244.95):
Thus, Jill is indifferent about the partnership investment at a price of $11,848.47. At this price, Gary is better off forming the partnership rather than retaining a sole proprietorship, since the utility of the sole proprietorship is 250 whereas the utility of the partnership (at a price of $11,848.47) is 252.459.
It would also be interesting to explore the price at which Gary is indifferent about the sole proprietorship versus the partnership. If you recalculate the spreadsheet for finding that price (where utility for the sole proprietorship = utility for the partnership = 250), a price of $10,625 is implied. Since this price represents a substantial discount from Jill’s “breakeven” price of $11,848.47, she will enjoy utility of 247.461 versus utility of 244.95.