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:

Adverse Selection Class Problem and Solutions (from our October 4th class meeting)

Here’s the link to the Adverse Selection Class Problem that we worked on during our October 4th class meeting:

http://fin4335.garven.com/fall2018/Adverse_Selection_Dynamics_Numerical_Example

And here’s the link to the solutions for this class problem:

http://fin4335.garven.com/fall2018/Adverse_Selection_Dynamics_Numerical_Example__Solutions_

The spreadsheet which I used to produce the solutions document is available at http://fin4335.garven.com/fall2018/Dynamic%20adverse%20selection%20numerical%20example.xlsx.

Friendly reminder – Michael Munger extra credit opportunity tomorrow – 4-5:15 in Foster 250

Tomorrow, you can earn extra credit by attending and reporting on Dr. Michael Munger‘s lecture entitled “Tomorrow 3.0: Thriving in the the New Transaction Costs Economy” from 4-5:15 pm in Foster 250:


If you decide to take advantage of this opportunity, I will use the grade you earn 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 Dr. Munger’s lecture. In order to receive credit, the report must be submitted via email to risk@garven.com in either Word or PDF format by no later than Monday, October 15 at 5 p.m.

Grades report for Finance 4335, circa October 10, 2018

Earlier today, I updated Canvas to reflect course, attendance, quiz, problem set and exam grades for the 71 students who are currently enrolled in both sections of Finance 4335.  Here are the descriptive statistics for these grades.

Course Grade Attendance Quizzes Problem Sets Midterm Exam 1
Mean 84.22 95.21 70.90 86.04 83.56
Standard Deviation 11.55 6.39 13.82 12.61 20.47
Minimum 48.83 73.33 26.00 39.33 0.00
25th percentile 81.17 93.33 60.50 81.29 80.50
50th percentile 87.72 100.00 72.00 89.42 92.00
75th percentile 91.26 100.00 82.00 94.75 96.00
Maximum 98.83 100.00 95.00 100.00 100.00

Let me explain next how I calculated the “Course Grade”.  In the syllabus, the final (end-of-semester) course numeric grade calculation is given as

Final Course Numeric Grade = .10(Class Attendance) + .10(Quizzes) + .20(Problem Sets) + Max{.20(Midterm Exam 1) + .20(Midterm Exam 2) + .20(Final Exam), .20(Midterm Exam 1) + .40(Final Exam), .20(Midterm Exam 2) + .40(Final Exam)}

At the end of this semester, I will assign letter grades based upon the following schedule (which also appears in the course syllabus):

Final Course Letter Grade. The final course letter grade will be based upon the following schedule of final course numeric grades:

A 93-100% C 73-77%
A- 90-93% C- 70-73%
B+ 87-90% D+ 67-70%
B 83-87% D 63-67%
B- 80-83% D- 60-63%
C+ 77-80% F <60%

Since we are halfway through the semester, the equation for the Mid-Semester Course Numeric Grade which now appears in Canvas is calculated as :

Mid-Semester Course Numeric Grade =(.10(Class Attendance) + .10(Quizzes) + .20(Problem Sets) + .20(Midterm Exam 1)/.6

Keep in mind that you don’t actually receive a letter grade until the end of the semester.  However, in the meantime, this calculation at least provides a clear indication of how everyone is doing overall so far.

Adverse Selection – definition, examples, and some solutions

During yesterday’s Finance 4335 class meeting, I introduced the topic of adverse selection, and we’ll devote next Tuesday’s class to further discussion of this topic.

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.

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).

Case studies of how (poorly designed) insurance creates moral hazard

During yesterday’s class meeting, we discussed (among other things) how contract designs and pricing strategies can “fix” the moral hazard that insurance might otherwise create. Insurance is “good” to the extent that it enables firms and individuals to manage the risks that they face. However, we also saw insurance has a potential “dark side.” The dark side is that too much insurance and/or incorrectly priced insurance can create moral hazard by insulating firms and individuals from the financial consequences of their decision-making. Thus, in real world insurance markets, we commonly observe partial rather than full insurance coverage. Partial insurance ensures that policyholders have incentives to mitigate risk. Furthermore, real world insurance markets are characterized by pricing strategies such as loss-sensitive premiums (commonly referred to as “experience rated” premiums), as well as premiums that are contingent upon the extent to which policyholders invest in safety.

In competitively structured private insurance markets, we expect that the market price for insurance will (on average) be greater than or equal to its actuarially fair value. Under normal circumstances, one does not expect to observe negative premium loadings in the real world. Negative premium loadings are incompatible with the survival of a private insurance market, since this would imply that insurers are not able to cover capital costs and would, therefore, have incentives not to supply such a market.

Which brings us to the National Flood Insurance Program (NFIP). The NFIP is a federal government insurance program managed by the Federal Emergency Management Agency (also known as “FEMA”). According to Cato senior fellow Doug Bandow’s blog posting entitled “Congress against Budget Reform: Voting to Hike Subsidies for People Who Build in Flood Plains”,

“…the federal government keeps insurance premiums low for people who choose to build where they otherwise wouldn’t. The Congressional Research Service figured that the government charges about one-third of the market rate for flood insurance. The second cost is environmental: Washington essentially pays participants to build on environmentally-fragile lands that tend to flood.”

Thus, the NFIP provides us with a fascinating case study concerning how subsidized flood insurance exacerbates moral hazard (i.e., makes moral hazard even worse) rather than mitigates moral hazard. It does this by encouraging property owners to take risks (in this case, building on environmentally fragile lands that tend to flood) which they otherwise would not be inclined to take if they had to pay the full expected cost of such risks.

There are many other examples of moral hazard created by insurance subsidies. Consider the case of crop insurance provided to farmers by the U.S. Department of Agriculture.  The effective premium loading on federally provided crop insurance is typically quite negative (often in excess of -60%), thus putting crop insurance on a similar footing to flood insurance in terms of cost compared with actuarially fair value. Just as mis-priced flood insurance effectively encourages property owners to build in flood plains, mis-priced crop insurance incentivizes farmers to cultivate acreage which may not even be particularly fertile.

I could go on (probably for several hundred more pages – there are innumerable other egregious examples which I could cite), but I think I will stop for now…

Moral Hazard

During yesterday’s class meeting, we discussed the concept of moral hazard. In finance, the moral hazard problem is commonly referred to as the “agency” problem. Many, if not most real-world contracts involve two parties – a “principal” and an “agent”. Contracts formed by principals and agents also usually have two key features: 1) the principal delegates some decision-making authority to the agent and 2) the principal and agent decide upon the extent to which they share risk.

The principal has good reason to be concerned whether the agent is likely to take actions that may not be in her best interests. Consequently, the principal has strong incentives to monitor the agent’s actions. However, since it is costly to closely monitor and enforce contracts, some actions can be “hidden” from the principal in the sense that she is not willing to expend the resources necessary to discover them since the costs of discovery may exceed the benefits of obtaining this information. Thus, moral hazard is often described as a problem of “hidden action”.

Since it is not economically feasible to perfectly monitor all of the agent’s actions, the principal needs to be concerned about whether the agent’s incentives line up, or are compatible with the principal’s objectives. This concern quickly becomes reflected in the contract terms defining the formal relationship between the principal and the agent. A contract is said to be incentive compatible if it causes principal and agent incentives to coincide. In other words, actions taken by the agent usually also benefit the principal. In practice, contracts typically scale agent compensation to the benefit received by the principal. Thus, in insurance markets, insurers are not willing to offer full coverage contracts; instead, they offer partial insurance coverage which exposes policyholders to some of the risk that they wish to transfer. In turn, partial coverage reinforces incentives for policyholders to prevent/mitigate loss.

Similarly, in a completely different setting, consider the principal/agent relationship which exists between the owner and manager of a business. If the manager’s effort level is high, then the owner may earn higher profits compared with when the manager’s effort level is low. However, if managerial pay consists of a fixed salary and lacks any form of incentive compensation (e.g., bonuses based upon meeting or beating specific earnings targets), then the manager may be inclined to not exert extra effort, which results in less corporate profit. Thus, compensation contracts can be made more incentive compatible by including performance-based pay in addition to a fixed salary. This way, the owner and manager are both better off because incentives are better aligned.

Finance 4335