Problem Set 6 is due at the beginning of class on Tuesday, October 22.
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/fall2019/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).
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I just had a question about the problem set if you have the time. For part e, how are we supposed to know what the premium loading factor is? Because in the spreadsheet example that was used in the calculations but i don’t see how you are supposed to find that number in the problem set.
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The price of insurance (AKA the “insurance premium”) is , where corresponds to the expected value of the indemnity, and corresponds to the premium loading factor. When = 0, then ; i.e., insurance is actuarially fair. As I pointed out during class last Thursday, > 0 in most private insurance market settings; i.e., insurance is typically actuarially unfair because firms incur transaction costs in forming and managing risk pools. It follows (from the insurance pricing equation given above) that ; i.e., the premium loading factor represents the percentage markup over and above the actuarially fair premium.
The first step in solving Part E (as noted in today’s Problem Set 5 Helpful Hint) requires determining the state-contingent indemnity schedules () for the four risk management strategies (i.e., self-insurance, full insurance, deductible insurance, and coinsurance). These indemnity schedules represent the contractually agreed upon payments made by the insurer to the consumer under each contract in each of the three loss states. Once you’ve figured out the state-contingent indemnities for each policy type, then calculate the expected values for each indemnity schedule (i.e., ). Since the insurance prices (premiums) are $3,125 for Policy A and $2,500 for both Policy B and Policy C, then finding the premium loading factor for each policy is straightforward.
The logic required for solving part E of Problem Set 5 is virtually identical to the logic behind the Coinsurance, Deductibles, and Upper Limits Spreadsheet spreadsheet (shown in class last Thursday); i.e., you need to determine: 1) the indemnity schedules under four alternative risk management strategies (i.e., self-insurance, full insurance, deductible insurance, and coinsurance), 2) state-contingent wealth associated with these strategies, and 3) state-contingent utilities and expected utility for these strategies.
On September 26 and October 1, we’ll focus our attention on the first midterm exam in Finance 4335. Class on Thursday, September 26 will be devoted to a review session for the exam, and the exam will be administered during class on Tuesday, October 1.
A student asked me whether it is possible for the value of in problem set 4 to be negative. Here, represents the optimal level of exposure to the risky asset; 1- represents the optimal level of exposure to the safe bond. While it is certainly theoretically possible for to have a negative value, for this particular problem it turns out that > 0. The reason is positive in this case is that 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 , this means that you must have made a math error somewhere. In the case of investor A, E(U(W)) = .6(105 + 25 ).5 + .4(105 – 35 ).5, then one can find the optimal value for investor A’s by applying the chain rule individually to both the .6(105 + 25 ).5 and the .4(105 – 35 ).5 terms, setting the resulting equation equal to zero (this is the so-called “first order condition”) and solving for . Obviously the same logic applies to solving investor B’s problem; the only difference is that E(U(W)) = .6ln(105 + 25 ) + .4ln(105 – 35 ) for investor B.
This raises an interesting question; specifically, what would have to be different about this problem in order to obtain a negative value for ? If this problem had been originally parameterized such that the expected return on the stock was less than the expected return on the bond, this would guarantee a negative value for . For example, suppose that everything stayed the same, but that the state probabilities for the stock were 55/45 rather than 60/40. Note that with 55/45 state probabilities, the stock has an expected return of .55(.3) + .45 (-.3) = 3%, which is less than the guaranteed 5% return on the bond. Under this scenario, investor A’s optimal is -48.33%, which implies that she would optimally 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). On the other hand, investor B’s optimal is -24% when the state probabilities for the stock are 55/45 rather than 60/40; I will leave it as an exercise for the reader to determine the expected return and risk for investor B’s optimal portfolio under this alternative scenario.
As I noted in my previous Problem Set #4 hint, 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 . If you do this, be sure to email your spreadsheet model to firstname.lastname@example.org sometime prior to the beginning of next Thursday’s class meeting, at which time I’ll also collect your completed problem sets.
Problem set #4 (which is due on Thursday, September 26) consists of two problems: 1) an optimal (expected utility-maximizing) portfolio problem, and 2) a stochastic dominance problem. We’ll discuss stochastic dominance next Tuesday, but in the meantime allow me to provide you with some hints for setting up the first problem.
The first problem involves determining how to (optimally) allocate initial wealth W0 = $100 to (risky) stock and (safe) bond investments for two investors who are identical in all respects except utility. Let represent the allocation to stock; then the plan is to invest $100 in the stock and $100(1-) in the bond. The key here is to find the value for 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 = $100 for both investors;
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 for either investor, 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 = W0(1.3) + (1-)W0(1.05) = 100(1.30) + (1-)100(1.05) = 130 + (1-)105 = 105 + 25.
40% of the time, Ws = W0(.7) + (1-)W0(1.05) = 100(.7) + (1-)100(1.05) = 70 + (1-)105 = 105 – 35.
Therefore, expected utility for Investor A is: E(U(W)) = .6(105 + 25).5 + .4(105 – 35).5, and expected utility for Investor B is E(U(W)) = .6ln (105 + 25) + .4ln(105 – 35). It is up to you to solve for the optimal value of for each investor. There are two ways to do this – via calculus or a spreadsheet model. Actually, I would encourage y’all to work this problem both ways because doing so will help you develop an even better grasp of the underlying principles and concepts in Finance 4335. However, at your option, you may rely solely on building your own spreadsheet model. If you do this, in order to receive full credit, you need to email your spreadsheet model to email@example.com along with turning in the completed problem set.
Problem Set 2 consists of two problems. The first problem requires calculating expected value, standard deviation, and correlation, and using this information as inputs into determining expected returns and standard deviations for 2-asset portfolios; see pp. 15-18 of the http://fin4335.garven.com/fall2019/lecture3.pdf lecture note for coverage of this topic. The second problem involves using the standard normal probability distribution to calculate the probabilities of earning various levels of return by investing in risky securities and portfolios; see pp. 17-22 of the http://fin4335.garven.com/fall2019/lecture4.pdf lecture note for coverage of that topic.