Category Archives: Economics

Federal Financial Guarantees: Problems and Solutions

Besides insuring bank and thrift deposits, the federal government guarantees a number of other financial transactions, including farm credits, home mortgages, student loans, small business loans, pensions, and export credits (to name a few).

In order to better understand the problems faced by federal financial guarantee programs, consider the conditions which give rise to a well-functioning private insurance market. In private markets, insurers segregate policyholders with similar exposures to risk into separate risk classifications, or pools. As long as the risks of the policyholders are not significantly correlated (that is, all policyholders do not suffer a loss at the same time), pooling reduces the risk of the average loss through the operation of a statistical principle known as the “law of large numbers”. Consequently, an insurer can cover its costs by charging a premium that is roughly proportional to the average loss. Such a premium is said to be actuarially fair.

By limiting membership in a risk pool to policyholders with similar risk exposures, the tendency of higher risk individuals to seek membership in the pool (commonly referred to as adverse selection) is controlled. This makes participation in a risk pool financially attractive to its members. Although an individual with a high chance of loss must consequently pay a higher premium than someone with a low chance of loss, both will insure if they are averse to risk and premiums are actuarially fair. By charging risk-sensitive premiums and limiting coverage through policy provisions such as deductibles, the tendency of individuals to seek greater exposure to risk once they have become insured (commonly referred to as moral hazard) is also controlled.

In contrast, federal financial guarantees often exaggerate the problems of adverse selection and moral hazard. Premiums are typically based upon the average loss of a risk pool whose members’ risk exposures may vary greatly. This makes participation financially unattractive for low risk members who end up subsidizing high risk members if they remain in the pool. In order to prevent low risk members from leaving, the government’s typical response has been to make participation mandatory. However, various avenues exist by which low risk members can leave “mandatory” risk pools. For example, prior to the reorganization of the Federal Savings and Loan Insurance Corporation (FSLIC) as part of the Federal Deposit Insurance Corporation (FDIC) during the savings and loan crisis of the 1980s and 1990s, a number of low risk thrifts became commercial banks. This change in corporate structure enabled these firms to switch insurance coverage to the FDIC, which at the time charged substantially lower premiums than did the FSLIC. Similarly, terminations of overfunded defined benefit pension plans enable firms to redeploy excess pension assets as well as drop out of the pension insurance pool operated by the Pension Benefit Guarantee Corporation (PBGC).

Although financial restructuring makes it possible to leave mandatory insurance pools, the costs of leaving may be sufficiently high for some low risk firms that they will remain. Unfortunately, the only way risk-insensitive insurance can possibly become a “good deal” for remaining members is by increasing exposure to risk; for example, by increasing the riskiness of investments or financial leverage. Furthermore, this problem is even more severe for high risk members of the pool, especially if they are financially distressed. The owners of these firms are entitled to all of the benefits of risky activities, while the insurance mechanism (in conjunction with limited liability if the firm is incorporated) minimizes the extent to which they must bear costs. Consequently, it is tempting to “go for broke” by making very risky investments which have substantial downside risk as well as potential for upside gain. The costs of this largely insurance-induced moral hazard problem can be staggering, both for the firm and the economy as a whole.

Ultimately, the key to restoring the financial viability of deposit insurance and other similarly troubled federal financial guarantee programs is to institute reforms which engender lower adverse selection and moral hazard costs. Policymakers would do well to consider how private insurers, who cannot rely upon taxpayer-financed bailouts, resolve these problems. The most common private market solution typically involves some combination of risk-sensitive premiums and economically meaningful limits on coverage. Federal financial guarantee programs should be similarly designed so that excessively risky behavior is penalized rather than rewarded.

On the economics of financial guarantees

On pp. 13-19 of the Derivatives Theory, part 3 lecture note and in Problem Set 9, we study how credit enhancement of risky debt works. Examples of credit enhancement in the real world include federal deposit insurance, public and private bond insurance, pension insurance, mortgage insurance, government loan guarantees, etc.; the list goes on.

Most credit enhancement schemes work in the fashion described below. Creditors loan money to “risky” borrowers who are at risk for defaulting on promised payments. Although borrowers promise to pay back $B at t=1, they may default (in whole or in part) and the shortfall to creditors resembles a put option with t=1 payoff of -Max[0, B-F]. Therefore, without credit enhancement, the value of risky debt at t=0 is

V(D) = B{e^{ - r}} - V(Max[0,B - F]).

However, when credit risk is intermediated by a guarantor (e.g., an insurance company or government agency), credit risk gets transferred to the guarantor, who receives an upfront “premium” worth V(Max[0,B - F]) at t=0 in exchange for having to cover a shortfall of Max[0,B - F] that may occur at t=1. If all credit risk is transferred to the guarantor (as shown in the graphic provided below), then from the creditors’ perspective it is as if the borrowers have issued riskless debt. Therefore, creditors charge borrowers the riskless rate of interest and are paid back what was promised from two sources: 1) borrowers pay D = B - Max[0,B - F], and 2) the guarantor pays Max[0,B - F].


Rothschild-Stiglitz Numerical Example

In class last Thursday, while I explained the logic behind the so-called Rothschild-Stiglitz model (also presented in the last couple of paragraphs of my “Adverse Selection – a definition, some examples, and some solutions” posting and in pp. 22-23 of my “Moral Hazard and Adverse Selection” lecture note), I didn’t quite get around to providing a numerical example, so here it is.

Suppose that insurance clients are identical in all respects except for accident risks.  Specifically, all such clients have initial wealth of $125, U(W) = W^.5 (i.e., square root utility), and exposed to a loss of $100 when an accident occurs.  There are only two possible states – accident and no accident.  However, some clients are high risk (i.e., probability of accident is p(h) = .75, whereas other clients are low risk; i.e., p(l) = .25.

The insurer’s problem is that he/she cannot determine which clients are high risk and which clients are low risk; consequently, the insurer decides to implement the Rothschild-Stiglitz model so as to induce self selection by clients.  Specifically, a high risk contract called Policy A is offered which provides 100% coverage for $75, and a low risk contract called Policy B is offered which provides 10% coverage for $2.50.  The following spreadsheet shows that high risk clients will self-select into Policy A, thus earning 0 profit for the insurer, whereas low risk clients will self-select into Policy B, thus earning 0 profit for the insurer:

Here, note that high risk and low risk clients prefer policies A and B respectively.  Certainly the insurer would like for low risk clients to select Policy A (since this would provide profit to the insurer of $50 per client), but low risk clients rationally select Policy B since it maximizes their expected utilities.  Of course, the objective here is to make sure that Policy B is not attractive to high risk clients by limiting the level of coverage offered under this contract, and indeed, Policy A (which offers 100% coverage at a price that is actuarially fair to high risk clients) is preferred.  Since Policy B offers 10% coverage at a price that is actuarially fair to low risk clients, this high risk clients prefer Policy A; however, if the coinsurance rate on Policy B were, say, 20%, then high risk clients would prefer Policy B and there would be adverse selection (involving a per policy loss due to high risk clients opting into Policy B) of $10 per high risk client.

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

During yesterday’s Finance 4335 class meeting, I introduced the topic of adverse selection, and we’ll devote tomorrow’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.

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:


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.

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

Last week, 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 tomorrow’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.

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 earning 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.

Arrow-Pratt method vis–à–vis the “exact” method for calculating risk premiums

I received an email from a Finance 4335 student asking for further clarification of the two methods for calculating risk premiums which we covered in class last Thursday. Under the so-called “exact” method, one 1) calculates expected utility, 2) sets expected utility equal to the utility of the certainty-equivalent of wealth, 3) solves for the certainty-equivalent of wealth, and 4) obtains the risk premium by calculating the difference between expected wealth and the certainty-equivalent of wealth. A numerical example of this approach is provided on page 3 of the lecture note. On the other hand, the Arrow-Pratt method is an alternative method for calculating the risk premium which is based upon Taylor series approximations of expected utility of wealth and the utility of the certainty equivalent of wealth (the derivation for which appears on pp. 16-18 of this same lecture note). Both of these approaches for calculating risk premiums are perfectly acceptable for purposes of Finance 4335.

The value added of Arrow-Pratt is that it analytically demonstrates how risk premiums depend upon two factors: 1) the magnitude of the risk itself (as indicated by variance), and 2) the degree to which the decision-maker is risk averse. For example, we showed in class on Thursday that the Arrow-Pratt coefficient for the logarithmic investor (for whom U(W) = ln W) is twice as large as the Arrow-Pratt coefficient for the square root investor (for whom U(W) = W.5); 1/W for the logarithmic investor compared with .5/W for the square root investor. Thus, the logarithmic investor behaves in a more risk averse than the square root investor; other things equal, the logarithmic investor will prefer to allocate less of her wealth to risky assets and buy more insurance than the the square root investor. Another important insight yielded by Arrow-Pratt (at least for the utility functions considered so far in Finance 4335) is the notion of decreasing absolute risk aversion. Other things equal, investors become less (more) risk averse as wealth increases (decreases).

The Index Fund featured as one of “50 Things That Made the Modern Economy”

Tim Harford also features the index fund in his “Fifty Things That Made the Modern Economy” radio and podcast series. This 9 minute long podcast lays out the history of the development of the index fund in particular and the evolution of so-called of passive portfolio strategies in general. Much of the content of this podcast is sourced from Vanguard founder Jack Bogle’s September 2011 WSJ article entitled “How the Index Fund Was Born” (available at Here’s the description of this podcast:

“Warren Buffett is the world’s most successful investor. In a letter he wrote to his wife, advising her how to invest after he dies, he offers some clear advice: put almost everything into “a very low-cost S&P 500 index fund”. Index funds passively track the market as a whole by buying a little of everything, rather than trying to beat the market with clever stock picks – the kind of clever stock picks that Warren Buffett himself has been making for more than half a century. Index funds now seem completely natural. But as recently as 1976 they didn’t exist.  And, as Tim Harford explains, they have become very important indeed – and not only to Mrs Buffett.”

Warren Buffett is one of the world’s great investors. His advice? Invest in an index fund

Insurance featured as one of “50 Things That Made the Modern Economy”

From November 2016 through October 2017, Financial Times writer Tim Harford presented an economic history documentary radio and podcast series called 50 Things That Made the Modern Economy. This same information is available in book under the title “Fifty Inventions That Shaped the Modern Economy“. While I recommend listening to the entire series of podcasts (as well as reading the book), I would like to call your attention to Mr. Harford’s episode on the topic of insurance, which I link below. This 9-minute long podcast lays out the history of the development of the various institutions which exist today for the sharing and trading of risk, including markets for financial derivatives as well as for insurance.

“Legally and culturally, there’s a clear distinction between gambling and insurance. Economically, the difference is not so easy to see. Both the gambler and the insurer agree that money will change hands depending on what transpires in some unknowable future. Today the biggest insurance market of all – financial derivatives – blurs the line between insuring and gambling more than ever. Tim Harford tells the story of insurance; an idea as old as gambling but one which is fundamental to the way the modern economy works.”