Rothschild-Stiglitz model and related numerical example from last Thursday’s class meeting

This past Thursday, we discussed the logic behind the Rothschild-Stiglitz “separating equilibrium” model (see pp. 22-24 of the Moral Hazard and Adverse Selection lecture note) and provided a numerical illustration of its inner workings .

From our initial study of adverse selection in insurance markets (see pp. 17-21 of the Moral Hazard and Adverse Selection lecture note), we find that low-risk insureds cross-subsidize high-risk insureds when pooled premiums (based upon the average of the expected costs for both risk-types) are charged. In the “dynamic” version where there are many different risk-types (see Adverse Selection Dynamics Class Problem), this results in the so-called “insurance death spiral”. The death spiral begins with the exit of the lowest risk members of the pool, because pooling makes insurance too expensive for them and they are better off self-insuring. Their exit causes premiums for remaining pool members to increase, which motivates even more lower risk members to also exit, further shrinking the risk pool and making it even more expensive. Unchecked, this dynamic ultimately results in the failure of the insurance market.

The purpose of the static (2 risk-types) Rothschild-Stiglitz model is to show how insurance contract design can mitigate the adverse selection problem described in the previous paragraph. By offering full coverage contracts (based on the high-risk loss probability) and partial coverage (based on the low-risk probability), the high-risk and low-risk types credibly confirm whether they are high-risk or low-risk by their contract choices.  Since the full coverage contract provides high-risk types with greater expected utility than the partial coverage contract, and the partial coverage contract provides low-risk types with greater expected utility than the full coverage contract,  voilà – the adverse selection problem goes away because the insurer now knows who’s who!

The problem that we worked on toward the end of Thursday’s class meeting provides a  numerical illustration of the Rothschild-Stiglitz model. Here is the problem description (from pg. 24 of the Moral Hazard and Adverse Selection lecture note):

Note that Policy A represents actuarially fair full coverage based on the high-risk probability, whereas Policy C represents actuarially fair partial coverage based on the high-risk probability.  Without any further calculation, the Bernoulli principle implies that high-risk types will prefer Policy A over Policy C, and that Policy A and Policy C are preferred to self-insurance.  Furthermore, Policy B will never be offered, since high-risk types prefer Policy B over A and the insurer would lose $19.50 ($65-$45.50) per high-risk type if it offered Policy B.

Since we are interested in determining the policy pair which maximizes (expected value of) profit, it all boils down to whether  the insurer offers Policy C or D.  We already know that the high-risk types prefer A over C.  We need to determine whether the low-risk types prefer C or D, and whether there’s any possibility that high-risk types might defect from A to D if D were offered (note that the choice of D over A by high-risk types loses money for the insurer, since the expected cost of 30% coverage of high-risk types costs $19.50, and policy D’s premium is only $13.65).  Furthermore, while we know that high-risk types prefer A to C, we don’t yet know  the preference ordering by low-risk types of self-insurance, Policy C, and Policy D.  Under Policy C, the expected profit per low-risk type is $39 – .6(35) = $18, but it is only $13.65-$10.50 = $3.15 under Policy D.

The following spreadsheet provides with the answers that we need (clicking on the picture below brings up the spreadsheet from which this picture is obtained; see the worksheet labeled as “RS (Class Problem)”):

The various calculations in this worksheet confirm our intuition – the profit maximizing pair is A and C.  If A and C is offered, then the insurer earns expected profit of $0 on A per high-risk type (because A is purchased exclusively by high-risk types) and expected profit of $18 on C (because C is purchased exclusively by low-risk types).  If Policy D is offered instead of Policy C, then high-risk types still prefer A (and yield expected profit per high-risk type of $0), whereas low-risk types prefer D (and yield expected profit per low-risk type of $3.15)

Moral Hazard and Adverse Selection Class Problems and Solutions

Here are the moral hazard and adverse selection class problems and solutions. These class problems were featured during the 10/15/2019 class meeting of Finance 4335.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty.  Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden.  For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.  

Plan B for those who cannot attend Catherine Semcer’s talk tonight

Here’s a Plan B for those who have schedule conflicts with Catherine Semcer‘s talk tonight entitled “Saving Africa’s Wildlife: The Failure of Hunting Bans and Success of Entrepreneurial Solutions”. It turns out that Ms. Semcer was a featured guest on an EconTalk podcast episode hosted by Russ Roberts this past February. I am willing to accept a written report on Ms. Semcer’s EconTalk episode in lieu of a written report on her presentation tonight at Baylor.

The same rules apply for the podcast as they do for tonight’s live presentation. If you decide to take advantage of this podcast extra credit opportunity, I will use the grade you earn on your report to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest quiz grade). The report should be in the form of a 1-2 page executive summary in which you provide a critical analysis of Russ Robert’s 67 minute interview with Ms. Semcer entitled “Poaching, Preserves, and African Wildlife”. In order to receive credit, the report must be submitted via email to fin4335in either Word or PDF format by no later than Monday, October 21 at 5 p.m.

Problem Set 6 Hints and Spreadsheet

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).

Adverse Selection – a definition, some examples, and some solutions

During today’s Finance 4335 class meeting, we will delve deeper into the topic of adverse selection. Adverse selection is often referred to as the “hidden information” problem. This concept is particularly easy to understand in an insurance market setting; if you are an insurer, you have to be concerned that the worst possible risks are the ones that want to purchase insurance. However, it is important to note that adverse selection occurs in many market settings other than insurance markets. Adverse selection occurs whenever one party to a contract has superior information compared with his or her counter-party. When this occurs, often the party with the information advantage is tempted to take advantage of the uninformed party.

In an insurance setting, adverse selection is an issue whenever insurers know less about the actual risk characteristics of their policyholders than the policyholders themselves. In lending markets, banks have limited information about their clients’ willingness and ability to pay back on their loan commitments. In the used car market, the seller of a used car has more information about the car that is for sale than potential buyers. In the labor market, employers typically know less than the worker does about his or her abilities. In product markets, the product’s manufacturer often knows more about product failure rates than the consumer, and so forth…

The problem with adverse selection is that if left unchecked, it can undermine the ability of firms and consumers to enter into contractual relationships, and in extreme cases, may even give rise to so-called market failures. For example, in the used car market, since the seller has more information than the buyer about the condition of the vehicle, the buyer cannot help but be naturally suspicious concerning product quality. Consequently, he or she may not be willing to pay as much for the car as it is worth (assuming that it is not a “lemon”). Similarly, insurers may be reticent about selling policies to bad risks, banks may be worried about loaning money to poor credit risks, employers may be concerned about hiring poor quality workers, consumers may be worried about buying poor quality products, and so on…

A number of different strategies exist for mitigating adverse selection. In financial services markets, risk classification represents an important strategy. The reason insurers and banks want to know your credit score is because consumers with bad credit not only often lack the willingness and ability to pay their debts, but they also tend to have more accidents than consumers with good credit. Signaling is used in various settings; for example, one solution to the “lemons” problem in the market for used cars is for the seller to “signal” by providing credible third party certification; e.g., by paying for Carfax reports or vehicle inspections by an independent third party. Students “signal” their quality by selecting a high-quality university (e.g., like Baylor! :-)). Here the university provides potential employers with credible third-party certification concerning the quality of human capital. In product markets, if a manufacturer provides a long-term warranty, this may indicate that quality is better than average.

Sometimes it’s not possible to fully mitigate adverse selection via the methods described above. Thus, insurers commonly employ pricing and contract design strategies which incentivize policyholders to reveal their actual risk characteristics according to their contract choices. Thus, we obtain what’s commonly referred to as a “separating” (Rothschild-Stiglitz) equilibrium in which high-risk insureds select full coverage “high-risk” contracts whereas low-risk insureds select partial coverage “low risk” contracts:

Rothschild-Stiglitz

The Rothschild-Stiglitz equilibrium cleverly restricts the menu of available choices in such a way that the insurer induces self-selection. Here, the insurer offers contract L, which involves partial coverage at an actuarially fair price (based upon the loss probability of the low risk insured), and contract H, which provides full coverage at an actuarially fair price (based upon the loss probability of the high risk insured). The differences in the shapes of the indifference curves are due to the different accident probabilities, with a lower accident probability resulting in a more steeply sloped indifference curve. Here, the high-risk policyholder optimally chooses contract H and the low-risk policyholder optimally chooses contract L. The high-risk policyholder prefers H to L because L would represent a point of intersection with a marginally lower indifference curve (here, the Ih curve lies slightly above contract L, which implies that contract H provides the high-risk policyholder with higher expected utility than contract L). The low-risk policyholder will prefer L, but would prefer a full coverage contract at the point of intersection of APl line with the full insurance (45 degrees) line. However, such a contract is not offered since both the low and high-risk policyholders would choose it, and this would cause the insurer to lose money. Thus, one of the inefficiencies related to adverse selection is that insurance opportunities available to low-risk policyholders are limited compared with the world where there is no adverse selection.

There is a very practical implication of this model. If you are a good risk, then you owe it to yourself to select high-deductible insurance, since insurers price low-deductible insurance with the expectation that high-risk policyholders will be the primary purchasers of such coverage (and therefore, low-deductible policies will be more costly per dollar of coverage than high-deductible policies.

Mark your calendars – Finance 4335 extra credit opportunity!

I have decided to offer the following extra credit opportunity for Finance 4335. You can earn extra credit by attending and reporting on Catherine Semcer‘s upcoming talk entitled “Saving Africa’s Wildlife: The Failure of Hunting Bans and Success of Entrepreneurial Solutions”.  Ms. Semcer’s talk is scheduled for Thursday, October 17 from 5:15-6:30 p.m. in Foster 250.

If you decide to take advantage of this opportunity, I will use the grade you earn on your report to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest quiz grade). The report should be in the form of a 1-2 page executive summary in which you provide a critical analysis of Ms. Semcer’s lecture. In order to receive credit, the report must be submitted via email to fin4335@gmail.com in either Word or PDF format by no later than Monday, October 21 at 5 p.m.

 

Problem Set 5 Solutions are available

Problem Set 5 Solutions are available for the exclusive use of students who are currently enrolled in Finance 4335 during the Fall 2019 semester at Baylor University.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty.  Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden.  For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.  

Finance 4335