By Cather, David A

INTRODUCTION One of the largest and least understood industries in the world is the insurance industry. Browne et al. (2000) estimate that 8 percent of the world’s GDP is spent on insurance products. Global insurance premiums totaled $3.7 trillion in 2006, and the United States ranked first among all countries as the largest consumer of insurance products in the world (Swiss Re, 2007). For most noncommercial consumers in the United States, insurance consumption has focused on personal insurance products, such as auto, health, homeowners, life, and retirement insurance products. But while they spend a great deal of money on insurance, many consumers appear to have a limited understanding of the factors that affect their demand for insurance products.1

People buy insurance for a variety of reasons. In some instances, the purchase is required by law, as is the case for compulsory auto liability insurance. In other cases, consumers buy insurance to satisfy the requirements of other parties that they are doing business with; e.g., mortgage lenders require that homeowners buy insurance on the homes that the lenders are financing as a condition of obtaining loans. Many others buy insurance due to a more abstract reason, a behavioral tendency known as risk aversion. As it relates to the demand for insurance, risk aversion refers to the tendency to prefer to pay a defined sum of money that is known with certainty instead of being exposed to the risk of suffering a larger and uncertain financial loss in the future. Thus, the practice of individuals buying insurance protection instead of bearing risk without insurance reflects risk aversion.

Scholarly texts typically explain the concept of risk aversion using an economic model called utility theory. Readers with previous course work in economics can find excellent discussions on utility theory, risk aversion, and the demand for insurance in several different books.2 Readers without such course work may find it difficult to understand utility theory, however, as discussions on the topic are usually written assuming that readers have a fairly advanced understanding of economics. Moreover, unless they are particularly familiar with the insurance industry, readers often find it difficult to relate the concepts discussed in most descriptions of utility theory to practical applications in their insurance consumption decisions.

The objective of this article is to provide a general introduction to utility theory and risk aversion as it applies to the demand for insurance. This primer thus provides an intermediate reading that covers the prerequisites assumed in more advanced discussions on the topic. To demonstrate that utility theory is not an abstract “ivory tower” exercise with limited relevance to the real world, the article includes several examples that show how risk aversion relates to real-life insurance demand patterns. In the next section, we describe how many consumers exhibit behavioral tendencies toward wealth that lend themselves to risk-averse behavior. The middle section explains how risk aversion results in a demand for insurance protection and examines how this demand is affected by insurance adirtinistrative loading charges. The final section of the article demonstrates how utility theory applies to everyday insurance purchasing decisions, and describes anecdotal evidence on how risk aversion has shaped consumer demand patterns in the personal auto insurance markets.

THE UTILITY THEORY MODEL

The relationship between risk aversion and the demand for insurance can be introduced by developing a simple model of consumer preferences regarding wealth. The model is built around two assumptions. First, it is assumed that consumers always prefer more wealth instead of less wealth. Second, it is assumed that as a person’s wealth increases, the amount of additional satisfaction generated by each additional dollar in his or her possession will decrease. It is easy to visualize these assumptions by representing them pictorially. A graph that reflects these two assumptions is shown by the darkened curve in Figure 1, where personal wealth is plotted along the horizontal axis, and the satisfaction associated with wealth is shown on the vertical axis. On the horizontal axis, wealth is measured in dollars (or any other measure of currency). The vertical axis is measured in terms of utility, a numeric scale that can be used to measure the level of satisfaction associated with a given wealth level. Along the vertical axis, larger values of utility reflect higher levels of satisfaction.3 Thus, as circumstances prompt a person to move from one wealth level to another, we can compare the utility levels corresponding to both wealth levels to determine which is preferred and how much utility has changed.

Utility, Measured as the Square Root of Wealth

The darkened curve shown in Figure 1 is drawn to reflect the above two assumptions. Thus, if we are risk averse, the curve reflecting our utility level will continually increase as our wealth increases (i.e., as we move from left to right along the curve), in keeping with the first assumption. Increased wealth is thus associated with increased satisfaction, an assumption that few rational people find problematic. The utility generated by each additional dollar is not constant across different wealth levels, however, as indicated by Assumption 2. Instead, we see that a $1 increase in wealth is much more important to us when we are poor than when we are wealthy, as that additional dollar generates a much larger amount of additional utility at lower wealth levels. In short, for an increase in wealth of an additional dollar, we receive much more “bang for our buck” when are poor than when we are rich.

Can we identify a simple way to calculate the utility derived from wealth that is consistent with the above assumptions? In fact, a variety of functional forms are available, but one of the simplest is to measure utility as the square root of wealth. As shown in Figure 1, this function results in a curve that continuously increases at a decreasing rate for wealth levels greater than or equal to zero, consistent with our two assumptions.4 One can thus calculate the utility associated with a given level of wealth by finding the square root of that wealth level. If a person’s wealth increases from $1 to $4 to $49, her corresponding utility increases from 1 to 2 to 7 units of utility, respectively. In keeping with these assumptions, it is thus easy to show that, as our wealth increases, we will receive an ever-decreasing amount of additional utility from each additional dollar of wealth.

Exercise 1: Calculating “Bang for the Buck.” Using a functional form that is consistent with the two assumptions discussed above, calculate the additional utility derived from increasing the wealth of a person from four dollars to five dollars, and compare it to the additional utility generated by increasing the wealth ofthat same person from $64 to $65.

Calculating utility as the square root of wealth, we can show how the additional utility that a person receives from possessing an additional dollar depends on that person’s initial wealth level, and that each additional dollar provides less additional utility as the person’s wealth increases.

* Receiving an additional dollar when we possess $4 increases our utility from 2 (the square root of $4) to 2.236 (the square root of $5), resulting in a 0.236 unit increase in utility.

* Receiving an additional dollar when we hold $64 increases our utility from 8.0 (the square root of $64) to 8.062 (the square root of $65), resulting in a 0.062 unit increase in utility.

These calculations show how risk-averse people value an additional dollar more when they are poor than when they are rich, as the utility generated by an additional dollar is higher when their initial wealth is $4 (0.236) than when their initial wealth is $64 (0.062). As shown in Figure 1, these figures measure the marginal utility corresponding to each wealth level, calculated as the slope of a line drawn tangent to the utility curve at wealth levels of $4 and $64, respectively.

The world is full of examples that suggest that the marginal utility of an additional dollar of wealth decreases as a person’s wealth increases. The cash-starved college student who rarely tipped at his favorite campus coffee shop often tips his servers more generously after obtaining a well-paying job. Many well-to-do professors who view the cost of $100 textbook as a reasonable and necessary expense were once students who complained bitterly about such prices. Wealthy philanthropists who started their careers with little money routinely make headlines for donating multimillion dollar gifts. As suggested by these and many other examples, these preferences regarding wealth are very prevalent in our economy today. In fact, these tendencies are so common that economists have coined a phrase for them, referring to them as the dimimshing marginal utility with respect to wealth.5

So far, we have focused on the relationship between utility and wealth, with no discussion on how this relationship relates to risk. We next consider how diminishing marginal utility creates a condition in which people exhibit an aversion to risk. Risk aversion refers to the behavioral tendency of preferring a less risky alternative instead of a choice with greater uncertainty when both choices have the same financial payoff over the long run. Risk aversion has been used to explain how consumers make choices in a variety of financially risky decisions, from investments to gambling. For this article, we will examine how risk aversion affects a person’s behavior when exposed to the risk of suffering a large financial loss that may be caused by a variety of adverse events, ranging from a loss of health to an auto accident. We will also explain why most individuals choose to buy insurance protection from such risks instead of bearing the risk uninsured. Expected Utility With No Insurance

Consider risk-averse Ray, a person who exhibits diminishing marginal utility with respect to wealth. The utility that Ray receives from wealth is measured as the square root of that wealth level.6 Ray’s initial wealth is equal to $64, but he is exposed to a possible loss equal to $60. We will assume that this loss is insurable, but we will keep our discussion general enough to encompass a variety of insurable risks, such as a sudden medical expense, a house fire, or a lawsuit. Ray is examining his options for dealing with his exposure to loss. Ray faces a 25 percent chance of suffering such a loss and a 75 percent chance of suffering no loss.

Note that Ray’s expected utility without insurance reflects a 25 percent probability of having utility equal to 2.0 (the square root of $4) if he suffers the loss and a 75 percent chance of having utility equal to 8.0 (the square root of $64) without a loss. We show the value for EUni on the vertical axis of the graph in Figure 1. Note that it corresponds to point E, a point on the dashed line that is one-fourth of the way down the line from point A toward point D, reflecting the 25 percent chance that the loss may occur.7

Expected Utility With Insurance Priced at the Pure Premium

When we compare Ray’s utility under the insurance option (7.0) to his expected utility without insurance (6.5), we see that Ray will have higher satisfaction if he buys insurance at the pure premium rate. We can also see this result in Figure 1. Paying $15 for insurance can be shown on the graph by starting at point A and moving left along the curve to point B.8

Why is insurance priced equal to the pure premium preferred? First, note that if Ray does not buy insurance, he faces a fairly high probability of suffering a catastrophic loss. A loss of this size results in Ray losing some dollars that he values quite highly, as we noted earlier in Exercise 1 when we found the additional utility from increasing our wealth from $4 to $5. By contrast, if Ray buys insurance, he is paying for the pure premium using low- marginal utility dollars, as we demonstrated in Exercise 1 when we found the additional utility from increasing our wealth from $64 to $65. Moreover, by buying insurance, Ray knows with certainty what his wealth level will be; he has transferred the risk of not knowing if his wealth will be $64 or $4 to the insurer in exchange for paying a fixed $15 pure premium.

In summary, risk-averse people will prefer to buy insurance priced at the pure premium rate rather than assume the risk of suffering a significant financial loss. This result follows from the fact that risk-averse people exhibit diminishing marginal utility with respect to wealth. As a result, they would prefer to buy insurance, paying with low-marginal utility dollars, instead of being exposed to the risk that they may suffer a large loss that would cost them high-marginal utility dollars.

While they would prefer to buy insurance priced at the pure premium rate, risk-averse people cannot typically buy such protection in real life, as insurers cannot make a practice of selling their products at such low prices without going out of business. Insurers must instead charge prices in excess of the pure premium rate to pay for the cost of their administrative expenses. The portion of a premium that is used to pay insurer adrrdnistrative expenses is referred to as the loading charge. These charges pay for such expenses as commissions to agents, compensation to insurer employees, underwriting expenses, taxes, claims administration expenses, and the general expenses common across all business enterprises. Loading charges vary across different types of insurance, generally ranging from 30 percent to over 50 percent of an insurer’s premium volume. Despite this wide range, the loading charges for personal lines insurance are fairly consistent, generally accounting for slightly less than 40 percent of the cost of the premiums charged by insurers.9

While industry-wide mean loading expenses are fairly similar across the different types of personal insurance products, these figures do not show how loading expenses can vary among insurers within a given product line. To demonstrate the range of expenses among insurers, the data in Table 1 show the size of the loading expenses reported by several leading insurance companies in 2005 for private passenger auto liability, private passenger physical damage, and homeowners insurance. Since all the insurers shown in Table 1 rank among the largest insurers in their respective markets, they are generally large enough to take advantage of the cost efficiencies associated with economies of scale, thus reducing their loading charges below those charged by many smaller insurers. Nonetheless, the size of the expense loadings incurred by these insurers varies widely, with some insurers’ loading expenses exceeding their more cost-efficient competitors by over 50 percent. Much of the variation in these costs can be attributed to differences in the ways that insurers distribute their products, as insurers that do not use agents or brokers to market their products save a sizable amount of money on items like agents’ commissions and brokerage fees.10 Nonetheless, even the most cost-efficient insurers spend over 20 percent of their premium revenue on loading expenses.

Insurance Company Underwriting Expenses Incurred by Product Line in 2005

Expected Utility With Insurance Priced Above the Pure Premium

Given the size of insurer loading expenses, we must expand our explanation of the demand for insurance to apply to conditions under which consumers are willing to pay insurance premiums that are large enough to cover both the pure premium and loading expenses. We can use Figure 1 to demonstrate how risk aversion can create a situation in which consumers may be willing to buy insurance at prices in excess of the pure premium. Recall that we were able to show the utility level that corresponds to buying insurance priced at the pure premium by moving along the darkened curve from point A to B. Now consider if an insurer included an additional $1 charge into the premium to help pay for its loading expense. The consumer’s resulting wealth level would plot on the curve slightly to the left of point B, and the associated utility level corresponding to his remaining wealth would equal the square root of $48, or 6.93. The consumer would clearly be willing to buy insurance at a price of the pure premium plus $1, as his corresponding utility is 6.93, which is greater than the 6.5 expected utility level he faces without insurance. In fact, the consumer is quite willing to pay a $2 or $3 loading charge above the pure premium, as the corresponding utility levels still remain higher than the uninsured utility level of 6.5. Described in this manner, as we add each additional dollar of loading charge to the pure premium, our remaining wealth after paying the premium (and its corresponding utility level) can be plotted on Figure 1 by moving along the curve from point B to the left.

Note that our risk-averse consumer is willing to pay several dollars more than the pure premium. This finding is consistent with real life, as no insurer could remain in business over the long run by selling their policies at the pure premium with no margin to cover their administrative expenses. It is also important to recognize, however, that the consumer’s willingness to pay for loading expenses is not limitless. Consumers will not be willing to pay an excessive amount for insurance, for their utility after buying such high-priced insurance would be less than a utility level equal to 6.5, their expected utility without insurance.

In fact, it is a simple process to determine the maximum premium that a consumer would be willing to pay for insurance.11 Recognizing that our consumer’s expected utility without insurance equals 6.5, Figure 1 shows that this utility level intersects with the utility curve at point C. Keeping in mind that the utility curve measures utility as the square root of wealth, point C thus corresponds to a wealth level of $42.25 (i.e., the square root of $42.25 yields the utility level of 6.5). We know that consumers are willing to pay a small amount in excess of the pure premium for loading (i.e., moving along the curve from B to C), but that the total premium cannot be so large as to reduce the consumer’s remaining wealth below $42.25. Thus, the maximum amount that a consumer is willing to pay for insurance is found by subtracting $42.25 (the monetary value associated with C) from $64, yielding a difference of $21.75. The $21.75 maximum premium consists of the pure premium cost equal to $15, and a willingness by consumers to pay as much as $6.75 in loading expenses. Or, stated in other words, consumers will be indifferent between bearing the risk without insurance and buying insurance for $21.75. But if an insurer can sell its policy with a loading charge that is less than $6.75, consumers can increase their level of utility by buying insurance protection.

As this discussion suggests, consumers are more likely to buy insurance from insurers with low loading expenses than those with higher loading charges, as lower loading expense charges usually result in a lower overall cost of insurance. In fact, note that our risk-averse consumer is willing to pay a loading expense of up to 31 percent ($6.75/$21.75) of the premium, a margin that is more than enough to cover the loading expenses of many of the insurers shown in Table I.12 UTILITY THEORY: A REAL-LIFE EXAMPLE OF RISK AVERSION IN AUTO INSURANCE

As we have shown so far, the concept of diininishing marginal utility with respect to wealth provides a simple and compact explanation for why many consumers choose to buy insurance. Readers are often skeptical, however, about how relevant utility theory is in explaining how insurers operate. In truth, our model of insurance demand bears little resemblance to how insurers set their prices. Insurers do not hire psychologists to estimate how risk averse their customers are, nor do they attempt to forecast the maximum amount of loading that they can charge. Instead, most insurers set their premium rates based on the expected value of the losses to be paid plus a loading charge to cover their administrative expenses and a reasonable profit. While most insurers would welcome an opportunity to charge higher prices, competition among insurers typically prevents them from charging excessive loading charges to their customers. Thus, while consumers may be sufficiently risk averse that they are willing to pay a large amount to transfer their risk to an insurer, they also understand that they can use competition among insurers as a tool to reduce their insurance premiums well below the maximum price that they would be willing to pay for protection (e.g., by getting competing price quotes from different insurers).

While competition serves as a potent force in holding down the prices charged by insurers, we can nonetheless see evidence of consumer risk aversion in many everyday insurance purchase decisions. A common situation in which risk aversion is evident pertains to how people buy personal auto insurance. Two standard types of coverage included in nearly all personal auto insurance policies are liability coverage and physical damage coverage. Physical damage coverage pays the owner of a vehicle for the cost to repair or replace the vehicle if it is damaged in an accident or by a variety of other types of losses.13 Liability coverage protects the insured driver if he or she is sued or found legally responsible for the injuries suffered by an injured party due to an auto accident. Most states require drivers to buy auto liability insurance to assure that they have a financing source with which to compensate auto accident victims who are injured by the drivers’ negligence. By contrast, physical damage coverage is not compulsory, although providers of auto loans typically require the borrower/ auto buyer to buy physical damage coverage to protect the lenders’ loan interest until the loan has been paid off. Thus, if they have paid off their car loans or purchased their vehicles using cash, vehicle owners have the option to drop auto physical damage protection from their insurance coverage.

Comparisons of the size of the maximum losses associated with auto liability and auto physical damage reveal a dramatically different impact on most consumers. The maximum size of the loss paid under physical damage coverage is the depreciated market value of the covered vehicle. Given the rapid depreciation in value of most vehicles, older model vehicles are often worth no more than a few thousand dollars. By contrast, the financial cost of auto liability claims can be potentially catastrophic to a driver. If a driver severely injures a fellow motorist in an auto accident, the victim may incur tens or hundreds of thousands of dollars in medical expenses, physical therapy, lost wages, and similar types of charges. Additionally, if a victim successfully sues the negligent driver, the courts often award additional compensation for pain and suffering and similar types of nonmonetary compensation. Thus, in comparison to physical damage losses, auto liability awards represent a much more costly potential economic loss to most consumers.

The difference in the maximum size of loss between auto physical damage and auto liability results in significantly different levels of consumer demand for the two coverages, as is demonstrated in the following exercise.

Exercise 2: Applying the lessons of risk aversion to the demand for auto insurance. Otto has recently finished his MBA degree and is saving money to buy his first home. Otto’s current net worth is $60,900. In addition to his savings, he owns a 4-year old car, nicknamed Rusty, worth $10,000. Otto estimates that he has a 5 percent chance of causing an accident that could result in a $50,000 liability award. He anticipates that such an accident would cause sufficient damage to Rusty that the car would be totally destroyed. Assume that Otto’s utility with respect to wealth (i.e., the relationship between Otto’s wealth level and his utility derived from wealth) can be modeled as Utility = Wealth05. Using the methods described earlier in this article, answer the following questions regarding Otto’s demand for auto liability coverage ANO for auto physical damage coverage:

1. Calculate the pure premium required to cover the risk.

2. Determine if Otto will have higher utility by buying insurance priced at the pure premium rate or by absorbing the risk without insurance.

3. Determine the maximum amount that Otto is willing to spend to insure the risk. Or, stated in other words, what is the maximum loading expense that Otto is willing to pay in addition to the cost of the pure premium?

The calculations shown in Table 2 provide numerical solutions for the above three questions. The chief difference between the two types of coverage is the severity of the loss, as the liability loss is much larger than the physical damage loss of the car.14 By comparing the answers in the second and third columns, the table allows us to see how the answers pertaining to the demand for liability coverage differ from the answers for physical damage coverage. Thus, the figures shown in the first row indicate that the pure premium for liability insurance coverage ($2,500) is much larger than for physical damage coverage ($500), reflecting the higher severity of liability losses. By comparing the figures in second and third rows of each column, one can also see that Otto will choose to buy insurance priced at the pure premium rate for both types of coverage, since the utility of buying liability (physical damage) insurance at such a price is 241.66 (245.7641), as compared to the 239.66 (245.7208) level of utility without insurance.

The combined impact of suffering both the $10,000 property damage loss and the $50,000 liability loss is shown in the figures in the right-most column of Table 2, reflecting the worst-case loss scenario described in the example. In keeping with the results shown in the second and third columns, Otto achieves his highest level of utility by buying “pure premium” insurance protection for the combined $60,000 loss. More specifically, Otto’s utility is higher (240.6242) when he pays the combined pure premium of $3,000 to buy both liability and physical damage insurance protection than when he bears the risk of the $60,000 loss without insurance (utility equal to 235.9403).

Perhaps the most interesting comparison shown in Table 2 is the maximum price that Otto is willing to pay for protection from the liability and the physical damage losses, as shown in the bottom row of the table. Referring back to Figure 1, the maximum amount that a risk-averse person is willing to pay for insurance can be found by subtracting the monetary value associated with point C from the person’s initial wealth. Based on such calculations, Otto is willing to pay up to $3,462 for liability insurance, an additional $962 more than the $2,500 pure premium needed to cover the expected loss. This extra $962 provides a sizable margin with which to pay for insurer loading expenses.15 By contrast, Otto is willing to pay up to $521.29 for physical damage coverage on Rusty, an amount that builds in a loading charge that is less than 5 percent ([$521.29 - $500] / $521.29) of Otto’s maximum premium level. This premium provides an insufficient loading expense margin because even the most cost- effective insurers would require higher operating margins to conduct business in this market. As a result, it is unlikely that this risk will be insured, as a financially acceptable contract could not exist that satisfies both Otto and an auto physical damage insurer. Otto’s reluctance to pay a sufficient margin for loading on the physical damage loss reflects the loss’ minimal impact on his overall satisfaction, as the probability that his modest $10,000 car could be destroyed results in a comparatively small loss of utility.

Data Pertaining the Decision to Buy Auto Liability and Physical Damage Insurance

In practice, we find that insurance markets reflect the demand patterns described in Exercise 2. We can gain some insight about how much risk-averse drivers value liability insurance by exarruning the market for catastrophic liability coverage.16 Reflecting concerns about their exposure to huge lawsuits from auto accidents and other personal liability exposures, a growing number of individuals have purchased catastrophic liability insurance such as personal umbrella liability protection in recent years, and financial advisors routinely recommend such high-limit protection for high-wealth individuals.17 On the other hand, it is quite common for consumers to elect not to buy auto physical damage coverage on older cars as the vehicles’ depreciated value decreases with time. The destruction of an older, less valuable car, and the minhnal loss of utility associated with such a loss, poses a comparatively modest cost upon most drivers. Consumers are thus unwilling to pay much above the pure premium price for such protection. Industry statistics provide anecdotal support for this reasoning, as a leading auto insurer reports that about one-third of their new customers elect not to buy auto physical damage coverage.18 As these examples suggest, consumers are making purchasing decisions regarding different types of auto insurance protection that are consistent with utility theory, as buyers are less willing to bear loading expenses for smaller losses like physical damage losses than for potentially catastrophic losses like lawsuits. SUMMARY

Risk aversion is potent force fueling the demand for personal insurance. This article has provided an introduction to utility theory and risk aversion, explc^ining how diminishing marginal utility creates conditions that prompt consumers to demand insurance protection from financial loss.19 To demonstrate that utility theory has great practical impact on real-life insurance demand, the article shows how risk aversion creates a willingness by consumers to bear the cost of a reasonable, but not excessive, charge for insurance loading expenses and compares these charges across a cross- section of personal insurance products. The article also demonstrates how the influence of risk aversion can be seen in insurance demand patterns in the personal auto insurance market, as many people elect to drop physical damage protection because such losses pose a rather modest loss of utility to all but the least wealthy drivers, while a greater fraction of drivers buy insurance to protect themselves from potentially catastrophic liability losses.

Risk aversion is a behavioral phenomenon that cannot be easily measured. While we have simply measured utility as the square root of wealth throughout this article, one should recognize that risk aversion is not linked solely to this functional form and that the conclusions of this article hold true for a variety of other mathematical functions.20 Likewise, the article has not addressed many of the additional considerations (e.g., supply constraints, affordability issues) that can affect insurance demand. Instead, the purpose of this article has been to introduce risk aversion to readers unfamiliar with the concept so that they can better appreciate its impact on insurance demand. It is also hoped that this introduction of utility theory provides readers with a basic understanding of the concept so that they will feel better prepared to read more advanced discussions on the topic.21 Toward this goal, readers who wish to make sure that they have mastered the introductory concepts of this article are encouraged to complete the practice problems shown below.

1. Bubba’s utility (U) preferences with respect to wealth (W) can be described by the utility function U = W^sup 0.5^. Assume that Bubba’s current wealth level equals $256. Also assume that Bubba faces a 20 percent chance of incurring a fire in the next year that would totally destroy his home (valued at $155) and an 80 percent chance that no fire loss will occur.

A. Calculate the level of additional satisfaction that Bubba gets when his wealth increases from $1 to $2.

B. Calculate the level of additional satisfaction that Bubba gets when his wealth increases from $256 to $257.

C. Compare A and B above and explain why A provides Bubba more satisfaction.

D. Assume that an insurer was going to sell insurance for the above risk. How much is the pure premium for this risk? In your answer, explain what the pure premium measures.

E. Calculate Bubba’s level of satisfaction if he buys insurance for a price equal to the pure premium.

F. Calculate Bubba’s expected level of satisfaction if he assumes the risk of suffering the fire loss without buying insurance.

G. Compare E to F. Will Bubba buy insurance or assume the risk?

H. Assume that the insurer sells insurance for a price equal to $2 above the pure premium. Will Bubba buy insurance or assume the risk?

I. Determine the price of insurance that will leave Bubba indifferent between buying insurance and assuming the risk.

2. Refer to Exercises 1 and 2 in the article. Redo the calculations, assuming that utility equals the natural logarithm of the wealth level (i.e., U(x) = ln(x)).

3. Based on the results from the previous question, which function (the square root function or the natural logarithm function) reflects a greater amount of risk aversion?

INSTRUCTIONAL NOTES FOR “A GENTLE INTRODUCTION TO RISK AVERSION AND EXPECTED UTILITY”

Risk aversion is widely recognized as a leading reason why consumers demand insurance protection from financial loss. Tracing back to early work by von Neumann and Morgenstern (1944), the concepts of risk aversion and utility theory have assumed a prominent role in the academic literature on risk and insurance. Reflecting their importance in the literature, these concepts have traditionally been discussed in most introductory textbooks in risk and insurance, as well as related disciplines such as health economics. Curiously, however, an editorial shift has recently taken place in textbooks, as the discussions of risk aversion found in many newer editions of risk and insurance textbooks are quite limited. In fact, a review of the leading principles textbooks in the field of risk and insurance that have been published over the past 5 years reveals that most of these textbooks do not fist the term “risk aversion” or related terms in their index or their table of contents, or provide only a cursory description of the concept.

To address this omission, this article is designed to serve as a self-contained introduction to risk aversion as it applies to the demand for insurance. As a primer, the article is written for an audience that has limited background in economics or statistics and thus can be assigned to students with a diverse set of prerequisite skills. The article can be used in a principles class as the sole reading on the topic of risk aversion or as an introductory reading for more advanced discussions on the topic. Recognizing that many readers find it challenging to relate the abstract nature of risk aversion to real life, a key goal of the article is to provide practical examples of how risk aversion influences actual consumer demand patterns in the personal insurance markets.

INSTRUCTIONAL CHALLENGES AND SOLUTIONS

One possible explanation for the decreased coverage of risk aversion in many textbooks relates to the prerequisite skill set needed by students to understand of the topic. The traditional description of utility theory and risk aversion tends to be written for a readership with advanced course work in economics.22 Unfortunately, students majoring in business or other noneconomics specializations often do not have sufficient economics background to easily understand this material and therefore encounter difficulty in grasping the concept. Rather than cover these prerequisites, instructors often elect to omit a discussion of risk aversion from their syllabi. In turn, it appears that many textbooks have followed suit.

Given this dilemma, one of the most challenging aspects of covering risk aversion and utility theory in the classroom is finding a way to discuss the topic in a “student-friendly” manner that accommodates an audience with a diverse academic background. Some common trouble spots for students reading traditional discussions of risk aversion and utility theory include the four areas described below. The methods used in this article to address these issues are discussed below as well.

1. Lack of familiarity with notation. Many students are uncomfortable using the traditional notation that is found in most descriptions of utility theory (e.g., using references like U(Wx) or U(X) to describe the utility of $X of wealth). In recognition of this fact, this primer limits the use of such terminology, inserting specific numeric values for the value of X until the student becomes accustomed to such notation. Footnotes are also inserted at key points in the article to clarify areas pertaining to notation or technical details that frequently cause problems for students (e.g., see footnotes 3-8).

2. Functional form. To prevent a student’s lack of familiarity with utility models from interfering with learning, the article uses a very student-friendly functional form (i.e., the square root function for positive wealth levels) to model the utility curve. By asking students to calculate the numeric value of utility associated with different wealth levels using the square root function, students quickly grow to understand how that functional form is consistent with the traditional concave shape of the utility curve. (By contrast, students often fail to see how a model is consistent with dimimshing marginal utility with respect wealth if the vertical axis is not specified in numeric values or if the numbers shown on the vertical axis are assigned without a corresponding function that they can use to calculate such values, two practices commonly found in most discussions on the topic.) The wealth levels used in the beginning examples are selected carefully to assure that a student can do the math without a calculator, allowing the reader to concentrate on understanding the model instead of focusing on the calculations. Examples presented later in the article use more realistic wealth levels. To show that risk-averse behavior can be modeled using functional forms other than the square root function, the article includes practice problems that use the natural logarithm function to model utility.

3. Real-life examples. Students often resist abstract concepts like utility theory, claiming that they lack practical use in real- world situations. To a certain degree, students should not be criticized for this view, as most descriptions of utility theory fail to discuss how the model relates to real-life market situations. This shortcoming is unfortunate, as it is not difficult to integrate examples about how risk aversion relates to practical considerations of great importance to consumers, and the inclusion of such examples enables the skeptical student to see how utility theory is not simply an empty academic exercise with little relevance to the real world. In this regard, the primer addresses this problem in two ways. * In its discussion on why risk-averse consumers are willing to pay loading expenses in excess of the pure premium, the article provides actual insurance expense data (e.g., commissions, taxes, loss adjustment expenses) from leading personal lines insurers (see Table 1). This section provides a great opportunity to integrate practical topics like insurance company operations (e.g., the goals and costs of insurance underwriting, marketing, and claims administration) and regulation (state versus federal regulation, regulatory objectives, and the cost of taxes and assessments) into the discussion of risk aversion. Additionally, by providing data about the costs embedded in loading expenses, this section provides information that is useful in explaining the financial motivation behind the growth in self-insurance and alternative risk transfer mechanisms.

* In the third section, the article introduces anecdotal evidence about the actual demand patterns for two personal auto insurance coverages: personal auto liability and personal auto physical damage coverage. By comparing the potential loss of utility associated with severe liability claims to that for less costly collision and comprehensive claims, the article shows why consumers drop physical damage protection more frequently than protection from lawsuits. By relating the theoretical conclusions described in the risk aversion model to one of their first insurance purchases, students begin to understand the applicability of risk aversion to their real-life consumer decisions. The applications focus on personal auto insurance, as all students who are licensed drivers have a need for the coverage, and many students have at least some limited familiarity with the policy.

4. Numerical examples. Students gain confidence in their mastery of a concept when they can make practice calculations and verify that their numbers are correct. The primer includes two sets of exercises that can be used in class at key points in the lecture, allowing students to practice a key idea to see if they understand it correctly. Additionally, practice homework problems are provided at the end of the article to reinforce the numerical calculations covered in the article. Answers to the practice problems are available from the author upon request.

Copyright American Risk and Insurance Association, Inc. Spring 2010

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