Category Archives: Helpful Hints

Rothschild-Stiglitz model (numerical and graphical illustration)

According to the Rothschild-Stiglitz model that we studied during yesterday’s class, insurers will limit contract choices such that there is no adverse selection.  In the numerical example that we implemented during class, there are equal numbers of high risk and low risk insureds; all have initial wealth of $125 and square root utility.  There are two states of the world – loss and no loss, and the probabilities of loss are 75% for high risk types and 25% for low risk types.  By offering high risk types full coverage at their actually fair price of $75 and offering low risk types partial (10%) coverage at their actuarially fair price of $2.50, both types of risks buy insurance and there is no adverse selection.

This illustrated in the figure below and in the spreadsheet located at http://fin4335.garven.com/fall2018/rothschild-stiglitz-model.xls.  Clearly neither the B or C contracts would ever be offered because these contracts incentive high risks to adversely select agains the insurer.

Some observations concerning yesterday’s Rothschild-Stiglitz class problem

Yesterday, 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.

The objective here is to identify the set of contract offerings that would prevent adverse selection. If you consider the pricing of these four insurance contracts, Policy A involves full insurance that is actuarially fair for high-risk consumers. We know from the Bernoulli principle that these consumers would like to purchase this contract. The challenge is to identify contracts that are favorable for the low-risk consumers but not for the high-risk consumers. Clearly we would not want to offer contract B, since everyone would select this contract and we would lose $19.50 on every high-risk consumer who purchased it (while breaking even on every low-risk consumer). High-risk consumers won’t want Policy C because it offers actuarially fair partial coverage, which provides lower expected utility than actuarially fair full coverage. However, low-risk consumers would be willing to purchase Policy C, so if A and C were offered, the insurer would break even on A and make $18 in profit from low-risk consumers who purchase Policy C. Given a choice between being uninsured, buying Policy A, or buying Policy C, low-risk consumers would purchase Policy C since it would offer higher expected utility than the other alternatives. Policy D would also be an acceptable alternative; if high-risk consumers purchased this contract, the insurer would lose $5.85 per high-risk consumer. However, if Policy A was also offered, none of the high-risk consumers would purchase Policy D. But low-risk consumers would prefer Policy D since it would offer higher expected utility than the other alternatives.

For what it’s worth, I have uploaded a spreadsheet consisting of expected utility calculations for this problem:

Problem Set 6 Hints and Spreadsheet

I just posted a new problem set on the course website; specifically, Problem Set 6, which is due at the beginning of class on Thursday, October 11.

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/fall2018/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).

Midterm Exam 1 hints and formula sheet

I just posted the formula sheet for the exam at http://fin4335.garven.com/fall2018/formulas_part1.pdf. It is also linked as the first item on the formula sheets page on the course website.  This is identical to the formula sheet which will be attached to the exam booklet.

The exam consists of a total of 4 problems. The first problem is required, and you are also required to work 2 out of the 3 remaining problems on the exam (i.e., select two problems from Problems #2-#4). At your option, you may work all three of Problems #2-#4, in which case I will count the two problems with the highest scores toward your grade on this exam. Each of the graded problems is worth 32 points, so as a “bonus” I’ll add 4 points for including your name on the exam. Thus, the total number of points possible is 100.

Regarding content, the exam is all about stuff that we covered since the beginning of the semester; specifically, risk preferences, expected utility, certainty-equivalent of wealth, risk premiums, and stochastic dominance.

Guidelines for completing parts B and D on Problem Set 5

For parts B and D on Problem Set 5, you may solve these problems via either calculus or a spreadsheet model.

If you decide to implement a spreadsheet model, then you must email your spreadsheet model to “risk@garven.com” prior to the start of class on Tuesday. In the problem set that you turn in at the beginning of class on Thursday, please reference your spreadsheet when you explain your answers for parts B and D problem. However, if you rely upon the calculus for maximizing expected utility, then no spreadsheet is necessary, although you might consider validating the result that you obtain via calculus with a spreadsheet model anyway. Or, you could validate your spreadsheet model with the calculus.

In order to solve this problem via spreadsheet, you’ll need to use the so-called Solver Add-in. The instructions for loading the Solver add-in into Excel are provided at the following webpage:

https://support.office.com/en-us/article/load-the-solver-add-in-in-excel-612926fc-d53b-46b4-872c-e24772f078ca

Recap of Analytic and Numerical Proofs of the Bernoulli Principle and the Mossin Theorem

Yesterday, we discussed three particularly important insurance economics concepts: 1) the Bernoulli Principle,  2) the Mossin Theorem, and 3) the Arrow Theorem.  The Bernoulli Principle implies that if an actuarially fair full coverage insurance policy is offered, then an arbitrarily risk averse individual will purchase it. The Mossin Theorem implies that if insurance is actuarially unfair, then an arbitrarily risk averse individual will prefer partial insurance coverage over full insurance coverage.  Finally, the Arrow Theorem implies an arbitrarily risk averse individual will select deductible insurance from the menu of partial insurance choices that were considered during class yesterday.

In what follows, I present a recap of the analytic and numerical proofs of the Bernoulli Principle and Mossin Theorem which took up most of our attention in class yesterday.

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

Capture

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:

Solver1

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:

Solver2

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:

Solver3

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:

Solver4

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).

Hints for solving problem set #4 (Hint #1)

Problem set #4 consists of two problems: 1) an optimal (expected utility maximizing) portfolio problem, and 2) a stochastic dominance problem.  We’ll discuss stochastic dominance tomorrow (and also (hopefully) work a class problem in connection with that concept), 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 \alpha represent the allocation to stock; then the plan is to invest $100\alpha in the stock and $100(1-\alpha) in the bond. The key here is to find the value for \alpha 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
  • eEnd-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 = \alphaW0(1.3) + (1-\alpha)W0(1.05) = \alpha100(1.30) + (1-\alpha)100(1.05) = \alpha130 + (1-\alpha)105 = 105 + 25\alpha.
  • 40% of the time, Ws = \alphaW0(.7) + (1-\alpha)W0(1.05) = \alpha100(.7) + (1-\alpha)100(1.05) = \alpha70 + (1-\alpha)105 = 105 – 35\alpha.

Therefore, expected utility for Investor A is: E(U(W)) = .6(105 + 25\alpha).5 + .4(105 – 35\alpha).5, and expected utility for Investor B is E(U(W)) = .6ln (105 + 25\alpha) + .4ln(105 – 35\alpha). It is up to you to solve for the optimal value of \alpha 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 if time permits because doing so will help you develop an even better grasp of the underlying principles and concepts in Finance 4335.

Problem Set 2 helpful hints

Problem Set 2 is now available from the course website at http://fin4335.garven.com/fall2018/ps2.pdf; its due date is Tuesday, September 4.

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. 17-23 of the http://fin4335.garven.com/fall2018/lecture3.pdf lecture note for coverage of this topic. The second problem involves using the standard normal probability distribution to calculate probabilities of earning various levels of return by investing in risky securities and portfolios. We will devote tomorrow’s class meeting to these and related topics.

Problem Set 1 Hint…

Problem Set 1 is due at the beginning of class tomorrow. Here is a hint for solving the 4th question on this problem set.

The objective is to determine how big a hospital must be so that the cost per patient-day is minimized. We are not interested in minimizing the total cost of operating a hospital; a sure-fire way to minimize total cost would be to not even have a hospital in the first place. Indeed, if you were to differentiate the total cost function given by C = 4,700,000 + 0.00013X2 with respect to X, this is what the math would tell you.

In part “a” of the 4th question, you are asked to “derive” a formula for the relationship between cost per patient-day and the number of patient days; in other words, what you are interested in determining is what is the most cost-efficient way to scale a hospital facility such that the cost per patient-day is minimized. Once you obtain that equation, then you’ll be able to answer the question concerning optimal hospital size.

Lagrangian Multipliers

There is a section in the assigned “Optimization” reading tomorrow on pp. 74-76 entitled “Lagrangian Multipliers” which (as noted in footnote 9) may be skipped without loss of continuity.  The primary purpose of this chapter is to re-acquaint students with basic calculus and how to use the calculus to solve so-called optimization problems.  Since the course only requires solving unconstrained optimization problems, there’s no need for Lagrangian multipliers.

Besides reading the articles entitled “Optimization” and “How long does it take to double (triple/quadruple/n-tuple) your money?” in preparation for tomorrow’s meeting of Finance 4335, make sure that you fill out and email the student information form as a file attachment to risk@garven.com prior to the beginning of tomorrow’s class.  As I explained during yesterday’s class meeting, this assignment counts as a problem set, and your grade is 100 if you turn this assignment in on time (i.e., sometime prior to tomorrow’s class meeting) and 0 otherwise.