## Rothschild-Stiglitz model (numerical and graphical illustration)

According to the Rothschild-Stiglitz model (referenced in the Adverse Selection – a definition, some examples, and some solutions blog posting, and on page 22 of the Asymmetric Information: Moral Hazard and Adverse Selection lecture note), insurers will limit contract choices such that there is no adverse selection.  To see this, assume there are equal numbers of high-risk and low-risk insureds, all of whom have an 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 is illustrated in the figure below and in the spreadsheet located at http://fin4335.garven.com/spring2023/rothschild-stiglitz-model.xls.  Clearly, neither the B (full coverage for low-risk insureds) nor C (based on the average cost of the actuarially fair prices for the low-risk and high-risk) contracts would ever be offered because both of these contracts incentivize high-risk types to adversely select against the insurer.

Rothschild-Stiglitz model (numerical and graphical illustration)

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

The adverse selection problem (also known as the “hidden information” problem) is especially easy to grasp in an insurance market setting; if you are an insurer, you need to be concerned that the worst potential risks may be the ones who wish to purchase insurance. However, it is important to note that adverse selection also occurs in many other market settings. Adverse selection occurs whenever one party to a contract has superior information compared with his or her counterparty. When this occurs, there is a risk that the more informed party may take advantage of the other, less informed party.

In an insurance setting, adverse selection is an issue whenever insurers know less about the actual risk characteristics of a potential client than the client herself.  In lending markets, banks have limited information about their potential clients’ willingness and ability to pay back 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…

Several strategies exist for mitigating adverse selection. In financial services markets, risk classification represents an important strategy. Insurers and banks want to know your credit score because consumers with bad credit not only often lack the willingness and ability to pay their debts, but they also have more accidents on average 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., 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 signal that quality is better than average.

Sometimes it’s not possible to mitigate fully adverse selection via the methods described above. Thus, insurers commonly employ pricing and contract design strategies that financially reward policyholders for revealing their true risk characteristics according to the contract choices they make; i.e., they voluntarily reveal their preferences. Thus, we get 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 choices in such a way that the insurer induces self-selection; i.e., low-risk insureds select the (low-risk and profitable) partial coverage contract designed with them in mind, and the high-risk insureds select the (high-risk and profitable) full coverage contract designed for them.  Here, the insurer offers contract L, which involves partial coverage at an actuarially fair price (based on the loss probability of the low-risk insured), and contract H, which provides full coverage at an actuarially fair price (based on the loss probability of the high risk insured). The indifference curve slopes are steeper for the low-risk insureds than they are for the high-risk insureds.  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 prefers L to H, even though only provides partial coverage.   Thus, one inefficiency 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 expecting 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).

## Synopsis of the Moral Hazard topic…

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 that the agent may take actions that are not 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 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 risks 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.

## 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 economics, an important topic that we covered briefly during last Thursday’s meeting of Finance 4335.

Your Tolerance for Investment Risk Is Probably Not What You Think

## VIX is back in the news (Page 1 feature article in today’s WSJ)!

It’s back!  At the beginning of this semester, I introduced our class to the CBOE’s Implied Volatility Index (VIX) in my blog posting entitled “On the relationship between the S&P 500 and the CBOE Volatility Index (VIX)“.  In that posting, I pointed out how over relatively short time intervals, percentage changes in VIX and SP500 indices move inversely.

VIX measures market expectations for stock market (S&P500) volatility over the coming 30 days.  It is commonly referred to as a “fear index”, and as such, it is indicative of the near-term degree of overall investor risk aversion.

This article mostly focuses on how investor fears of more aggressive Fed rate hikes and a possible recession are causing prices of options to be bid up, as investors “scurry for protection”.

Investors Are Bracing for Surge in Market Volatility
Bets on a rise in Wall Street’s fear gauge swell to most since March 2020

## Actuarially Fair Value For an Insurance Policy

A Finance 4335 student asked me the following question via email earlier today:

Q: “How do you find the actuarially fair value for an insurance policy?”

Here’s my answer to this question:

A: The actuarially fair value corresponds to the expected value of the insurance indemnity; the indemnity is the amount of coverage offered by an insurance policy. Under “full coverage”, 100% of the loss is indemnified, and in such a case, the actuarially fair premium is equal to the expected value of the loss distribution.

The concept of “actuarially fair” insurance prices/premiums, along with implications for the demand for insurance, is explained in two previously assigned readings (italics added for emphasis):,

1. on page 4 of the Supply of Insurance assigned reading (just prior to the section entitled “Example 2: Correlated Identically Distributed Losses), the following sentence appears, “A premium that is equal to the expected outcome is called an actuarially fair premium”;
2. on page 30 of the Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2) assigned reading, consider the following excerpt: “Ignoring transaction costs, an insurer charging a premium equal to expected loss would break even if it held a large portfolio of such policies. This premium could be called a fair premium or an actuarially fair premium, denoting that the premium is equal to the expected value of loss (sometimes called the actuarial value of the policy). The term fair is not construed in a normative sense; rather it is simply a reference point”; and
3. on page 43 of the Basic Economics: How Individuals Deal with Risk (Doherty, Chapter 2) assigned reading, in the first sentence of the first full paragraph: “We know from the Bernoulli principle that a risk averter will choose to fully insure at an actuarially fair premium.”

## Some important intuitions from today’s class meeting of Finance 4335

1. The most important concept covered in class today a is that people vary in terms of their preferences for bearing risk. Although we focused most of our attention on modeling risk-averse behavior, we also considered examples of risk neutrality (where you only care about expected wealth and are indifferent about the riskiness of wealth) and risk loving (where you actually prefer to bear risk and are willing to pay money for the opportunity to do so).
2. Related to point 1: irrespective of whether you are risk averse, risk neutral, or risk loving, the foundation for decision-making under conditions of risk and uncertainty is expected utility. Given a choice among various risky alternatives, one selects the choice which has the highest utility ranking.
3. If you are risk averse, then $E(W) > {W_{CE}}$ and the difference between $E(W)$ and ${W_{CE}}$ is equal to the risk premium $\lambda$. Some practical implications — if you are risk averse, then you are okay with buying “expensive” insurance at a price that exceeds the expected value of payment provided by the insurer, since (other things equal) you’d prefer to transfer risk to someone else if it’s not too expensive to do so. On the other hand, you are not willing to pay more than the certainty equivalent for a bet on a sporting event or a game of chance.
4. If you are risk neutral, then $E(W) = {W_{CE}}$ and $\lambda = 0$; risk is inconsequential and all you care about is maximizing the expected value of wealth.
5. If you are risk loving, then $E(W) < {W_{CE}}$ and $\lambda < 0$; you are quite willing to pay for the opportunity to (on average) lose money.

## On the ancient origin of the word “algorithm”

The January 24th assigned reading entitled “The New Religion of Risk Management” (by Peter Bernstein, March-April 1996 issue of Harvard Business Review) provides a succinct synopsis of the same author’s 1996 book entitled “Against the Gods: The Remarkable Story of Risk“. Here’s a fascinating quote from page 33 of “Against the Gods” which explains the ancient origin of the word “algorithm”:

“The earliest known work in Arabic arithmetic was written by al­Khowarizmi, a mathematician who lived around 825, some four hun­dred years before Fibonacci. Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al­Khowarizmi” fast. That’s where we get the word “algo­rithm,” which means rules for computing.”

Note: The book cover shown above is a copy of a 1633 oil-on-canvas painting by the Dutch Golden Age painter Rembrandt van Rijn.

## On the relationship between the S&P 500 and the CBOE Volatility Index (VIX)

Besides going over the course syllabus during the first day of class on Tuesday, January 17, we will also discuss a particularly important “real world” example of financial risk. Specifically, we will study the relationship between realized daily stock market returns (as measured by daily percentage changes in the SP500 stock market index) and changes in forward-looking investor expectations of stock market volatility (as indicated by daily percentage changes in the CBOE Volatility Index (VIX)):
As indicated by this graph (which also appears in the lecture note for the first day of class), daily percentage changes on closing prices for the SP500 (the y-axis variable) and for the VIX (the x-axis variable) are strongly negatively correlated with each other. The blue dots are based on 8,315 contemporaneous observations of daily returns for both variables, spanning the 33-year period of time starting on January 2, 1990 and ending on December 30, 2022. When we fit a regression line through this scatter diagram, we obtain the following equation:

${R_{SP500}} = .00062 - .1147{R_{VIX}}$,

where ${R_{SP500}}$ corresponds to the daily return on the SP500 index and ${R_{VIX}}$ corresponds to the daily return on the VIX index. The slope of this line (-0.1147) indicates that on average, daily closing SP500 returns are inversely related to daily closing VIX returns.  Furthermore, nearly half of the variation in the stock market return during this time period (specifically, 48.87%) can be statistically “explained” by changes in volatility, and the correlation between ${R_{SP500}}$ and ${R_{VIX}}$ came out to -0.70. While a correlation of -0.70 does not imply that daily closing values for ${R_{SP500}}$ and ${R_{VIX}}$ always move in opposite directions, it does suggest that this will be the case more often than not. Indeed, closing daily values recorded for ${R_{SP500}}$ and ${R_{VIX}}$ during this period moved inversely 78.59% of the time.