Category Archives: Economics

Insurance death spiral in the news…

It turns out that the so-called “insurance death spiral” which we modeled in class this past Tuesday (see page 21 of the Moral Hazard and Adverse Selection lecture note for a verbal description and the Dynamic Adverse Selection Spreadsheet for a numerical illustration of the problem) is very much in the news these days; e.g., see the editorial entitled Salvaging Private Health Insurance in today’s Wall Street Journal and yesterday’s page 1 WSJ article entitled In Start to Unwinding the Health Law, Trump to Ease Insurance Rules.  Quoting from today’s WSJ editorial,

“ObamaCare’s defenders are calling all of this “sabotage” and warning about “adverse selection,” in which a more robust individual market will siphon off the healthy customers that prop up ObamaCare’s exchanges. They predict a death spiral of higher premiums for the sick or elderly left on the exchanges.”

Sound familiar?  As we discussed in class last Tuesday, since the implementation of combined premium schemes effectively force good risks to pay too much whereas bad risks pay too little, the good risks opt out.  When this occurs,  expected cost of claims (and correspondingly higher premiums) are in store for those who remain in the risk pool.

Some observations concerning the Rothschild-Stiglitz numerical example from today’s class…

Today, 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 4 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.

Here’s a spreadsheet consisting of expected utility calculations:


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

During last Thursday’s Finance 4335 class meeting, I introduced 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 forth…

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 a “separating” (AKA Rothschild-Stiglitz) equilibrium in which high-risk insureds select full coverage “high-risk” contracts whereas low-risk insureds select partial coverage “low risk” contracts:


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, you owe it to yourself to select high-deductible insurance. The problem with a low deductible is that you will unnecessarily bear adverse selection costs if you follow this strategy.

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

During last week’s class meetings, we discussed 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 still 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 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 (AKA “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) that they otherwise might not take if they had to pay the full expected cost of these 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. According to this Bloomberg article, the effective premium loading on federally provided crop insurance is more than -60%, thus putting crop insurance on a similar footing to flood insurance (in terms of its cost compared to its actuarially fair value). Once again, incorrect pricing encourages moral hazard. As the Bloomberg article notes,

“…subsidies give farmers an incentive to buy “Cadillac” policies that over-insure their holdings and drive up costs. Some policies protect as much as 85 percent of a farm’s average yield.”

Just as mis-priced flood insurance effectively encourages property owners to build in flood plains, mis-priced crop insurance incentivizes farmers to cultivate acreage that may or may not even be fertile.

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

Analytic and Numerical Proofs of the Bernoulli Principle and Mossin’s Theorem

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.

The Bernoulli Principle is graphically illustrated in 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 \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:


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:


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:


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:


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).

Moral Hazard

During next Tuesday’s class meeting, we will discuss 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.

On the Determinants of Risk Aversion

In January 2014, The Economist published a particularly interesting article about the determinants of risk aversion, entitled “Risk off: Why some people are more cautious with their finances than others”. Here are some key takeaways from this article:

1. Economists have long known that people are risk-averse; yet the willingness to run risks varies enormously among individuals and over time.

2. Genetics explains a third of the difference in risk-taking; e.g., a Swedish study of twins finds that identical twins had “… a closer propensity to invest in shares” than fraternal ones.

3. Upbringing, environment and experience also matter; e.g., . “…the educated and the rich are more daring financially. So are men, but apparently not for genetic reasons”.

4. People’s financial history has a strong impact on their taste for risk; e.g., “… people who experienced high (low) returns on the stock market earlier in life were, years later, likelier to report a higher (lower) tolerance for risk, to own (not own) shares and to invest a bigger (smaller) slice of their assets in shares.”

5. “Exposure to economic turmoil appears to dampen people’s appetite for risk irrespective of their personal financial losses.” Furthermore, a low tolerance for risk is linked to past emotional trauma.

More on the St. Petersburg Paradox…

During last Thursday’s class meeting, we discussed (among other things) the famous St. Petersburg Paradox. The source for this is Daniel Bernoulli’s famous article entitled “Exposition of a new theory on the measurement of risk”. As was the standard academic practice in academia at the time, Bernoulli’s article was originally published in Latin in 1738. It was subsequently translated into English in 1954 and published a second time that same year in Econometrica (Volume 22, No. 1): pp. 22–36. Considering that this article was published 277 years ago in an obscure (presumably peer reviewed) academic journal, it is fairly succinct and surprisingly easy to read. The direct link to the PDF of this article is here: (P.S.: the article is gated but you can access it from anywhere on campus when connected to Airbear).

Also, the Wikipedia article about Bernoulli’s article is worth reading. It provides the mathematics for determining the price at which the apostle Paul would have been indifferent about taking the apostle Peter up on this bet. Specifically, if Paul were a millionaire, he should be willing to pay up to $20.88, On the other hand, if Paul were a thousandaire (total wealth = $1000), he should be willing to pay up to $10.95. Finally, if Paul was quite poor (e.g., total wealth = $2), then he might consider borrowing $1.35 and paying up to $3.35 to place this bet…

Your Tolerance for Investment Risk Is Probably Not What You Think

This WSJ article is authored by Professor Meir Statman,  the Glenn Klimek Professor of Finance at Santa Clara University.  Professor Statman’s research focuses on behavioral finance, which is a very important topic in decision theory that I hope to cover during tomorrow’s meeting of Finance 4335.
The questions financial advisers ask clients to get at the answer actually measure something completely different—often leading to misguided investment strategies.