Chapter 13 Economics of Insurance

13.1 Introduction

The objective of this chapter is to demonstrate how to use microeconomic tools (contract theory and uncertainty economics) to analyze and understand supply and demand behaviors in the insurance market. Historically, the microeconomic theory of insurance emerged in the mid-1960s from the collaboration between an actuary, Karl Borch, and an economist, Kenneth Arrow, who later received the Nobel Prize (or rather the “Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel” (not awarded by the Royal Swedish Academy of Sciences, but by a financial institutio)). Karl Borch aimed to provide a satisfactory theoretical foundation for insurance practices, while Kenneth Arrow sought an application field for uncertainty economics, a field of economics he had helped establish. Since Arrow and Borch, the convergence between actuarial science and microeconomic theory has been fruitful, to the extent that utility theory has become one of the methodological pillars of actuarial science.

The decision theory in uncertain environments allows for a relatively concise representation of various economic agent behaviors in the face of risk. It provides a precise meaning to concepts such as risk premium and risk aversion and models the choices of investors. The expected utility theory primarily aims to rationalize the choices of economic agents when confronted with risky situations. In this regard, throughout this chapter, we assume that the economic agent is concerned only with their final wealth level.

The significant developments in decision theory under uncertainty are closely associated with the names of Bernoulli, von Neumann, Morgenstern, and Savage. This theory has seen substantial development since then, with contributions from Arrow, Mossin, Elrich, or Becker, among others. Particularly, it is possible to describe agent behavior under uncertainty, or more generally under non-probabilistic uncertainty (sometimes referred to as ambiguity). Subsequently, research in this field has been extensive and has explored various directions, with one of the main focuses being the analysis of asymmetric information problems in the insurance market.

The first section will review the basics of decision models under uncertainty and the optimal risk-sharing (insurer/insured) in an exchange economy. The second part will continue Chapter 5 of Volume 1, focusing on risk aversion and risk measures. The third section will emphasize insurance supply and demand, as well as the form of optimal contracts (building on Arrow’s work). The fourth section will address equilibrium in the insurance market in the presence of asymmetric information (of any kind). Finally, the last section will economically approach the concepts introduced in Chapter 8 of Volume 1 regarding agent behavior in the presence of multiple risks. In particular, we will study risk coverage in the presence of a non-insurable component, as well as mechanisms for covering multiple risks (whether independent or not).

13.2 Decision under Uncertainty

Microeconomics is the branch of economics that examines the decision-making mechanisms of buyers and sellers. The products traded in the insurance market are risks. When an individual purchases an insurance policy, they exchange their own risk, for which they pay a premium, for another risk. This other risk may correspond to certain wealth (in the case of comprehensive insurance) or may contain a residual risk. Formally, by paying a premium $\pi(X)$, an individual exchanges risk $X$ for a payment of indemnity $I(X)$. Any financial risk can be associated with a real random variable, reducing the analysis of insurance behaviors to a decision problem under uncertainty that can be addressed using corresponding decision models.

13.2.1 The von Neumann and Morgenstern Expected Utility Model

13.2.1.1 Is Mathematical Expectation a Choice Criterion?

Let’s consider a decision maker facing two investments with financial returns represented by random variables $X$ and $Y$. The simplest way to choose between these two investments is to calculate the average returns and select the investment with the highest average return. To understand that individuals do not always behave as simple comparison of mathematical expectations would suggest, let’s examine the following example.

Example 13.1 Let’s consider the following two random wealths: $X =$ 10,000 and \[ Y=\left\{ \begin{array}{l} \text{-4,000 \euro with probability 0.3}\\ \text{18,000 \euro with probability 0.7.} \end{array} \right. \] If the decision maker chooses $X$, they receive 10,000 with certainty. In contrast, with $Y$, they face a loss of 4,000 in 30% of cases and a gain of 18,000 in 70% of cases. Since \[ \mathbb{E}[X]=10\,000<\mathbb{E}[Y]=11\,400, \] a decision maker who acts to maximize the mathematical expectation of their wealth should choose $Y$. However, the authors, like most readers, would choose $X$ because it presents less risk. Any individual declaring a preference for $X$ implicitly rejects the evaluation criterion based on mathematical expectation.

However, it’s essential to note an important aspect here: the temporal aspect. In this chapter, we are working with models involving two time periods. The economic agent makes a decision at time $0$, and we are interested in gains or losses at time $1$ (a one-period model). In particular, no dynamic aspect is considered here, and in the , a risky strategy can turn out to be more profitable without being risky.

Example 13.2 Thus, in Example $\ref{ExEspXY}$ above, the average gain over a large number of plays will converge (in probability) to 11,400 for $Y$, which exceeds the 10,000 provided consistently by $X$. In other words, in the long term, the law of large numbers guarantees us that the investment in $Y$ will be more profitable, even though it is riskier.

13.2.1.2 The Problem Posed by Nicholas Bernoulli

The choice criterion based on mathematical expectation has other disadvantages, as pointed out by Nicholas Bernoulli as early as 1738. Specifically, he posed the following problem to the Saint Petersburg Academy of Sciences:

Since the coin is fair, the probability of receiving $2^n$ is the same as getting heads $n-1$ times followed by one heads, which is $(1/2)^n$. The average gain associated with this game is therefore

\[ \sum_{n=1}^{+\infty}2^n\frac{1}{2^n}=+\infty, \]

so a decision-maker basing their decisions on mathematical expectation should be willing to pay an infinite sum to participate. However, it turned out that, even though the average gain is theoretically infinite, the maximum amount decision-makers are willing to pay to participate in this game is finite and even modest. This apparent paradox is known in the literature as the Saint Petersburg Paradox.

13.2.1.3 Solutions to the Paradox

Three solutions were proposed to this paradox by scholars of the time:

The first solution introduces the constraint of the wealth of the game’s organizer and considers any promise of payment exceeding this wealth as not credible. This amounts to limiting the player’s maximum gain by the financial capacity of the game’s organizer, making the average gain finite.
The second solution suggests considering events with probabilities below a certain threshold as impossible. However, the average gain then depends on the arbitrary choice of this threshold.
The third solution, proposed by Gabriel Cramer and Daniel Bernoulli and later axiomatized by Von Neumann and Morgenstern around the time of World War II, involves introducing a utility function for income representing the preferences of the economic agent. We will detail this approach below.

Remark. However, what has been mentioned so far does not question the value of mathematical expectation for actuaries. As we saw in Chapter 2, this mathematical tool allows, when the assumptions underlying the law of large numbers are satisfied, the calculation of the pure premium, the amount an insurance company must charge its clients to be able to compensate for claims falling under coverage without excess or deficit. This is explained by the context in which the insurance company operates, grouping a large number of similar and independent risks within a portfolio. Thus, the criterion of mathematical expectation is entirely valid for evaluating a random wealth as long as it is included in a large set of identically distributed independent random variables. When wealth must be considered in isolation, or when dependence arises, the criterion of mathematical expectation is no longer satisfactory.

13.2.1.4 The Principle of Bernoulli

Gabriel Cramer and Daniel Bernoulli proposed to resolve the Saint Petersburg Paradox by stating that decision-makers do not base their decisions on the average incomes of investments. Instead, the same gain can be perceived very differently based on the initial wealth of the decision-maker and certain traits of their character. In short, a gain of 1 euro is felt very differently by a poor soul with no money and a millionaire. This is why they introduced the concept of “moral value of money” as a choice criterion rather than the traditional mean value. Formally, every decision-maker is assumed to have a utility function $u$ such that the utility (or moral value) of wealth of x euros is given by $u(x)$.

A decision-maker choosing an investment among those modeled by the random variables $X$ and $Y$ will prefer the one that, on average, leads to the highest utility. Thus, given that the decision-maker’s initial wealth is $\omega$, they will agree to participate in the game described by Nicholas Bernoulli if the inequality \[\begin{equation} \label{IneqStPet} u(\omega)\leq \sum_{n=1}^{+\infty }u\left( \omega-p+2^{n}\right) \frac{1}{2^{n}} \end{equation}\] is satisfied, where $p$ represents the amount of the participation requested. Here, the individual compares the utility gained by participating in the game (right-hand side of $\eqref{IneqStPet}$) to the utility they would have by refusing to participate (left-hand side of $\eqref{IneqStPet}$). Inequality $\eqref{IneqStPet}$ expresses the fact that the decision-maker will participate in the game only if their average utility increases.

Cramer suggested that the utility $u(x)$ of wealth of $x$ euros is given by the square root $\sqrt{x}$ of that wealth. Bernoulli, on the other hand, suggested a logarithmic utility function $u(x)=\ln(x)$. For example, with an initial wealth of 10,000 euros and a logarithmic utility function, the decision-maker will only participate in the game if asked for less than 14.25 euros, even though the average gain is infinite. Note that the utility functions proposed by Cramer and Bernoulli are both increasing and concave. These two properties are fundamental, as we will see later.

Let’s conclude this section with a simple insurance example illustrating the importance of the utility concept for actuaries.

Example 13.3 Let’s consider an individual facing a risk of loss $S$ such that \[ S=\left\{ \begin{array}{l} 250,000 \text{ Euros, with probability }0.001,\\ 0 \text{ Euros, with probability }0.999. \end{array} \right. \] We can imagine that $S$ represents the consequences of a fire that would destroy the insured’s home with a probability of 0.001. In this case, $\mathbb{E}[S]=250 \text{ Euros}$. Most individuals would be willing to insure for a premium $p>\mathbb{E}[S]$ rather than risk losing 250,000 Euros. This clearly demonstrates that the utility of money is not linear. And it is precisely because policyholders are risk-averse that the insurance company can add a safety loading, thereby ensuring its solvency.

13.2.1.5 Axiomatization of the Expected Utility Model

We assume that each agent has a preference relation, denoted $$, in the set of real random variables, denoted $\mathcal{X}$. We will conventionally use $$ for indifference (i.e., $X\sim Y$ if and only if $X\preceq Y$ and $Y\preceq X$). In practice, it can be observed that this ordering relation can be defined over probability distributions, rather than random variables themselves. All the information is then contained in the knowledge of the cumulative distribution function $F_X$ of the random variable $X$. This is also called the assumption of neutrality, where an agent is neutral between two random variables that have the same cumulative distribution function. This notion corresponds to the first axiom of the expected utility model:

Axiom 1 The preference relation $\preceq$ depends only on the probability distribution of the random variables involved ($X\sim Y$ if $X=_{\text{law}} Y$), and we can write $X\preceq Y$ or $F_X\preceq F_Y$ interchangeably.

Now, let’s introduce the concept of a preference indicator.

Definition 13.1 A function $V$ : $\mathcal{X}\rightarrow {\mathbb{R}}$ represents the preference relation $$ if, and only if, for all $X,Y\in \mathcal{X}$, $X\preceq Y$ is equivalent to $V(X)\leq V(Y)$.

Before stating the axioms proposed by von Neumann and Morgenstern on the ordering relation, let’s recall that $F_{Z}$ is a convex combination of $F_{X}$ and $F_{Y}$ if there exists $\lambda \in [ 0,1]$ such that: \[\begin{equation*} F_{Z}(x)=\lambda F_{X}(x)+(1-\lambda )F_{Y}(x), \text{ for all } x. \end{equation*}\] Clearly, $F_Z$ is still a cumulative distribution function, corresponding to a discrete mixture of the probability distributions represented by the cumulative distribution functions $F_X$ and $F_Y$. The axioms of the expected utility model are as follows:

Axiom 2 The preference relation $\preceq$ is complete, reflexive, and transitive.
Axiom 3 For any cumulative distribution functions $F_{X}$, $F_{Y}$, and $F_{Z}$ that satisfy $F_{X}\preceq F_{Y}\preceq F_{Z}$, there exist $,$ $]0,1[$ such that: \[ \lambda F_{X}+(1-\lambda )F_{Z}\preceq F_{Y}\preceq \mu F_{X}+(1-\mu )F_{Z}. \]

Axiom 3 is often called the continuity axiom. It ensures that decision-makers’ preferences evolve smoothly without abrupt jumps. If $F_{X}\preceq F_{Y}\preceq F_{Z}$ is satisfied, this axiom assures us that this order will not change if we mix $F_X$, the worst-case scenario, with $F_Z$, the best-case scenario, in a certain proportion. Similarly, mixing $F_Z$ with $F_X$, in a certain proportion, does not alter the preference for that situation.

Axiom 4 For any cumulative distribution functions $F_{X}$, $F_{Y}$, and $F_{Z}$,

Axiom 4 is known as the axiom of independence. This axiom can be weakened (but remains equivalent under the assumption of neutrality) by considering the : an agent respects this principle if they maintain their preferences between two random variables that yield the same outcome for a given state of nature when that outcome is altered.

Axiom 5 Let $X$ and $Y$ be two constant variables such that $\Pr[X=x]=\Pr[Y=y]=1$, then $F_{X}\preceq F_{Y}$ implies $x\leq y$.

Independence is the fundamental axiom in the von Neumann-Morgenstern utility theory because it implies the possibility of finding a preference indicator, among all functions representing preferences, which takes the form of the mathematical expectation of the utility of decision outcomes. This function representing preferences is then linear with respect to probabilities and not necessarily linear with respect to outcomes. These axioms lead to a theorem known as the preference representation theorem in the expected utility model (see, for example, or for a proof).

Theorem 13.1 The preference relation $\preceq$ satisfies axioms 1-5 if and only if there exists a function $u: \mathbb{R} \rightarrow \mathbb{R}$, continuous, strictly increasing, and unique up to an increasing affine transformation, such that for any cumulative distribution functions $F_{X}$ and $F_{Y}$:

\[\begin{eqnarray*} F_{X}\preceq F_{Y}&\Leftrightarrow& \int_{\mathbb{R}}u(x)dF_{X}(x)\leq \int_{\mathbb{R}}u(x)dF_{Y}(x)\\&\Leftrightarrow& \mathbb{E}[u(X)]\leq \mathbb{E}[u(Y)]. \end{eqnarray*}\]

Preferences are then represented using the relation:

\[ V_{EU}(X)=\mathbb{E}[u(X)]. \]

The preferences induced by a decision-maker’s utility function are defined up to a positive linear transformation. Indeed, a decision-maker with a utility function $\widetilde{u}(\cdot)=au(\cdot)+b$ where $a>0$ would have exactly the same preferences as a decision-maker whose utility function is $u(\cdot)$. Thus, the measurement of utility is cardinal, akin to measures of distance, duration, or heat.

Remark. In particular, this allows us to standardize the utility function $u$ such that $u(0)=0$ and $u'(0)=1$. We can achieve this by replacing $u$ with $\widetilde{u}$ defined as:

\[ \widetilde{u}(x)=\frac{u(x)}{u'(0)}-\frac{u(0)}{u'(0)}. \]

13.2.2 On the Notion of Subjectivity in Probabilities

While the theory has been formulated assuming that agents know the probability of events, preferences in an uncertain non-probabilized context can be represented using the same type of criterion. The probability law that comes into play in this case is subjective. The choice set in this context consists of acts (mappings that associate an outcome with a state of nature). The axiomatics of the subjective expected utility model representing preferences in this context are due to , following the foundational work of Ramsey and de Finetti in the 1930s.

This subjectivity in the face of uncertainty means that agents remain rational but do not rely on probabilities. They form their own view of risk by introducing a subjective criterion. This has been observed, in particular, in the 1970s for events that are difficult to quantify, such as rare events (probability of a road accident, probability of having a trisomic child, probability of being infected with the HIV virus, etc.). Similarly, as demonstrated by , individuals generally tend to overestimate their ability to control events (the concept of “optimism bias”). For example, all automobile drivers have a high opinion of their driving skills and their ability to avoid accidents (see, for example, ).

13.3 The Role of Time and Prudence

A number of studies have also shown the need to incorporate the notion of time in thinking about risk aversion, induced by the concept of “prudence”. Especially for financial investments, a short time horizon appears to make agents more conservative.

notes that agents tend to take into account the probabilities of various possible events, but more importantly, they seek to protect themselves against those they perceive to be the most harmful. They seek to minimize the maximum possible loss. This situation can be modeled as the principle of . The agent then aims to avoid the worst outcome. In game theory, this strategy is called the strategy. Additionally, within this framework, a highly risk-averse agent no longer considers probabilities, but the assumption of rationality in their behavior is not rejected.

In the de Finetti-Savage approach, a representation theorem can be obtained by introducing a more general concept than utility, namely the notion of capacity (see Section $\ref{section-choquet}$).

13.3.1 Limits of the Expected Utility Model in Risk: Allais’ Paradox and the Certainty Effect

shows that agents can exhibit behavior contradictory to the expected utility model while still being “rational.” The origin of these behaviors is the violation of the independence axiom (and even more weakly, the sure-thing principle). Several empirical studies have subsequently confirmed the violation of this axiom.

Example 13.4 When asked to choose between the following two lotteries: \[ \begin{array}{ll} \text{Lottery A} & 100\% \text{ chance of receiving 1 million Euros,} \\ \text{Lottery B} & \left\{ \begin{array}{l} 10\% \text{ chance of receiving 5 million Euros,} \\ 89\% \text{ chance of receiving 1 million Euros,} \\ 1\% \text{ chance of receiving nothing,}% \end{array}% \right.% \end{array}% \] and between the following two lotteries: \[ \begin{array}{ll} \text{Lottery C} & \left\{ \begin{array}{l} 11\% \text{ chance of receiving 1 million Euros,} \\ 89\% \text{ chance of receiving nothing,}% \end{array}% \right. \\ \text{Lottery D} & \left\{ \begin{array}{l} 10\% \text{ chance of receiving 5 million Euros,} \\ 90\% \text{ chance of receiving nothing,}% \end{array}% \right.% \end{array}% \] people generally prefer $A$ to $B$ and $D$ to $C$, which violates the axioms of expected utility. In fact, a person preferring A to B has a utility function that satisfies: \[ u(1) \geq 0.1u(5) + 0.89u(1) + 0.01u(0), \] meaning that $0.11u(1) \geq 0.1u(5) + 0.01u(0)$. By adding $0.89u(0)$ to both sides, we obtain: \[ 0.11u(1) + 0.89u(0) \geq 0.1u(5) + 0.9u(0), \] indicating a preference for C over D. In this paradox, there is a (subjective) overweighting of certainty compared to an uncertain situation.

13.3.2 The Certainty Effect and Non-Linear Probability Processing

All of these experiments highlight a phenomenon known as the or (as named by Allais): the convex combination of two cumulative distribution functions with a third can reverse the order of preferences if the initially preferred cumulative distribution function corresponds to a degenerate variable (yielding the same outcome in all states of nature), and if after the mixture, the two cumulative distribution functions correspond to non-degenerate variables.

This behavior can be explained by the fact that the agent is sensitive to the fact that the certain choice guarantees a higher minimum gain than the risky choice, but if this is no longer the case after mixing, they turn to seeking the potential (expected) maximum gain. For such an agent, there is a significant psychological difference between a guaranteed minimum gain with a 100% probability and a minimum gain achieved with a probability close to 100%. This “overweighting” of certainty will be reflected in the model as a subjective transformation of probabilities, in addition to the subjective transformation of payments represented by the von Neumann and Morgenstern utility function.

13.3.3 Limits of the Expected Utility Model in Non-Probabilistic Uncertainty: The Ellsberg Paradox and the Concept of Ambiguity

Ellsberg’s work aimed to demonstrate that in real life, there are uncertain situations (see Section 2.2.2 for a definition of the concept of uncertainty) that cannot be treated as risky situations in the sense that agents cannot always assign subjective probabilities to events.

Example 13.5 Starting from an urn containing 300 balls, 100 red and 200 blue or green, consider the following lotteries: \[ \begin{array}{ll} \text{Lottery A:} & \text{win }1,000 \text{ Euros if the drawn ball is red,} \\ \text{Lottery B:} & \text{win }1,000 \text{ Euros if the drawn ball is blue,}% \end{array}% \] and the following two: \[ \begin{array}{ll} \text{Lottery C:} & \text{win }1,000 \text{ Euros if the drawn ball is not red,} \\ \text{Lottery D:} & \text{win }1,000 \text{ Euros if the drawn ball is not blue,}% \end{array}% \] people generally prefer $A$ to $B$ and $C$ to $D$, violating the axioms of expected utility. Preferring A to B and C to D means that: \[ \left\{ \begin{array}{l} \Pr[\text{red ball}] \geq \Pr[\text{blue ball}] \\ \Pr[\text{not red ball}] \geq \Pr[\text{not blue ball}],% \end{array}% \right. \] where $\Pr[\text{not red ball}] = 1 - \Pr[\text{red ball}]$ and similarly for the blue ball. These preferences seem to arise from the fact that betting on whether the ball will or will not be red is more “certain” than betting on whether the ball will or will not be blue.

13.3.4 Extension of the Notion of Expectation: Choquet Integral

As we saw (in Section 2.2.2 of Volume 1), a fundamental axiom for defining a probability measure was the property of additivity in the case of incompatible events, i.e., for two incompatible events, \[ \Pr [E \cup F] = \Pr [E] + \Pr [F]. \] This notion of probability then allowed us to define the fundamental concept of expectation (and pure premium, see Chapter 3). In particular, Property 3.2.7 ensures that the mathematical expectation of a risk $X$ can be represented as the integral over $\mathbb{R}^+$ of the associated tail function. This property extends to real random variables with finite expectation as follows: \[ \mathbb{E}[X] = -\int_{-\infty}^{0} \Pr [X \leq t] dt + \int_{0}^{\infty} \Pr [X > t] dt. \]% In 1957, Choquet proposed to define these notions in a more general framework, excluding the assumption of additivity, and defining not probabilities, but capacities.

Definition 13.2 Consider a probabilizable space $(\Omega,\mathcal{A})$. The set function $\nu: \mathcal{A} \rightarrow \mathbb{R}$ is a capacity if

$\nu(\emptyset) = 0$ and $\nu(\Omega) = 1$ (normalization property).
For all $A, B$ in $\mathcal{A}$, if $A \subset B$, then $\nu[A] \leq \nu[B]$.

This capacity will be called convex if and only if, for all $A$ and $B$ in $\mathcal{A}$, \[ \nu[A \cup B] + \nu[A \cap B] \leq \nu[A] + \nu[B]. \]% Finally, it will be called monotonically continuous if for any sequence $\{A_n, n = 1, 2, \ldots\}$ that is monotonic (in the inclusion sense) and $A_n \rightarrow A$, $\nu(A_n) \rightarrow \nu(A)$ as $n\to +\infty$.

We are now able to define the so-called Choquet integral.

Definition 13.3 Let $X$ be a random variable; the Choquet integral of $X$ with respect to the capacity $\nu$ is defined as \[ \int X d\nu = \int_{-\infty}^{0} \left(\nu[X > t] - 1\right) dt + \int_{0}^{\infty} \nu[X > t] dt. \]

Note that if $\nu$ is a probability, then the Choquet integral is simply the expectation of $X$. Capacities no longer satisfy the additivity properties; therefore, we deduce that the Choquet integral, unlike expectation, is not necessarily linear. However, we have the following result, which guarantees the additivity of the Choquet integral in the particular case where the random variables involved are comonotonic (this notion corresponds to perfect dependence; it was introduced in Section 8.2).

Proposition 13.1 If $X$ and $Y$ are two comonotonic random variables, then \[ \int (X + Y) d\nu = \int X d\nu + \int Y d\nu. \]

13.3.5 Generalization of Utility Expectation Models: Rank-Dependent Models

As we explained earlier, the perception of the likelihood of an event (and thus the associated probability) depends on the order of wealth in different states of nature. Thus, a change in a decision (e.g., through convex combination with another decision) that results in a change in the order of wealth across different states of nature also changes beliefs, which violates the axioms of utility expectation theory (independence and sure thing).

The axiom of independence is replaced by a weaker axiom.

Axiom 4’ Let $X$ and $Y$ be two discrete risks, taking values $\{x_1, \ldots, x_n\}$ and $\{y_1, \ldots, y_n\}$ with probabilities $p_i$, i.e., $p_i=\Pr[X=x_i]=\Pr[Y=y_i]$ for $i=1, \ldots, n$. We assume that $x_{1}\leq \ldots\leq x_{n}$ and $y_{1}\leq \ldots\leq y_{n}$, and that there exists $k_{0}$ such that $x_{k_{0}}=y_{k_{0}}$. If $X^{\prime }$ and $Y^{\prime }$ are obtained from $X$ and $Y$ by replacing $x_{k_{0}}=y_{k_{0}}$ with any other common value that preserves the same position, then $X\preceq Y \Leftrightarrow X^{\prime }\preceq Y^{\prime }$.

Under Axioms 1, 2, 3, and 4’, the corresponding representation of preferences is then \[\begin{equation*} V_{RDEU}(X)=\int u(x)d(g\circ \Pr)(x). \end{equation*}\] The function $g(\cdot )$ generally does not satisfy $g(p)+g(1-p)\neq 1$. It follows that the integral above is not a classical integral; it is a Choquet integral. In the case where $g(p)=p$, we recover the preference representation of the utility expectation model. This model is also sometimes called the dual model of Yaari.

It should be noted that this model was initially proposed by , who, instead of transforming probabilities, was the first to propose transforming the cumulative probabilities, i.e., the cumulative distribution function. This Rank-Dependent Expected Utility (RDEU) model integrates both a (non-linear) transformation of outcomes through the function $u$ and a (non-linear) transformation of probabilities through the function $g$. In his pioneering article, Quiggin mentioned overweighting of small probabilities and underweighting of large probabilities. For more details, see .

13.4 Risk Measure and Risk Aversion

13.4.1 Risk Aversion and Risk Price

Risk aversion can be seen from two perspectives: an economic agent can be considered risk-averse if they show a preference for certain gains over risky gains or if, between two random variables, they always prefer the less risky one. These two types of behaviors correspond, respectively, to weak risk aversion and strong risk aversion.

13.4.1.1 Weak Risk Aversion

The first concept of risk aversion is based on a preference for certain gains, with the reference being the risk premium (i.e., the mathematical expectation of the random variable). This type of behavior is called weak risk aversion.

Definition 13.4 A decision maker is said to be weakly risk-averse if, for any random variable, they prefer the variable that gives them its expected value with certainty.

In the utility expectation model, weak risk aversion corresponds to a concave utility function. The concepts of certainty equivalent and risk premium can be used to characterize risk aversion.

Definition 13.5 The certain equivalent of $X$ is the constant $c_{X}\in \mathbb{R}$ such that $c_{X}\sim X$. The risk premium of $X$ is the constant $\rho _{X}\in \mathbb{R}$ such that $\left(\mathbb{E}[X]-\rho_{X}\right) \sim X$.

The risk premium can be seen as the maximum amount that an agent is willing to pay to exchange the variable $X$ for its expected value. Moreover, it is easy to see that $\rho _{X}=\mathbb{E}[X]-c_{X}.$ The risk premium is a way to measure the intensity of weak risk aversion, allowing for the comparison of different agents’ degrees of risk aversion.

Remark. Before proceeding, it should be emphasized that risk aversion does not imply the outright rejection of all uncertainty. The fact that the risk-averse decision maker’s risk premium is always positive indicates that they are willing to pay to eliminate uncertainty. In the case of a financial asset, for example, a risk-averse decision maker will agree to sell the asset for a price lower than its expected return. However, the decision maker will retain the risky asset in their portfolio if the market does not offer them the price they demand to divest themselves of it. Therefore, it suffices for the mathematical expectation of the return to exceed the risk premium for the risk-averse decision maker to require payment to remove the asset from their portfolio. In the context of insurance, the risk-averse insured is willing to pay the risk premium in addition to the pure premium for risk coverage by the insurance company. Thus, there is nothing preventing a risk-averse decision maker from holding risky assets, provided that the return provided by them justifies the risk taken (i.e., the substitution of a portion of the constant wealth $\omega$ into risky assets).

13.4.1.2 Strong Risk Aversion

The concept of strong risk aversion first requires defining the notion of risk increase. However, there exist several concepts of risk increase, each corresponding to a specific definition of strong risk aversion. Variance as a measure of risk may seem intuitive, but in some cases, it leads to inconsistencies. To introduce a relatively general relationship, we will use the relationship $\preceq_{\text{TVaR}}$ introduced in Volume I.

Definition 13.6 A decision-maker is said to be strongly risk-averse if, for all random variables $X$ and $Y$ such that $X\preceq_{\text{TVaR}}Y$, they prefer $X$ over $Y$.

Rank-dependent utility models, which separate attitudes toward wealth from attitudes toward risk, also allow for distinguishing between weak and strong risk aversion. In these models, characterizations of these attitudes are based on conditions involving both the utility function and the probability distortion function: an agent is weakly risk-averse if and only if $u^{\prime \prime }<0$ and $g(p)\leq p$. In contrast, strong risk aversion corresponds to $u^{\prime \prime }<0$ and $g^{\prime \prime }>0$.

Comparing the risk behaviors of two agents is based on comparing their risk premiums. A local approximation of the risk premium reveals a risk aversion index, which is a criterion in comparing these behaviors (Arrow-Pratt theorem).

Proposition 13.2 Let $X$ be a random variable such that $X=+$ with $\omega\in{\mathbb{R}}$, $\mathbb{E}[\varepsilon]=0$, and $\mathbb{V}[\varepsilon]=\sigma ^{2}$. The decision-maker’s utility function is assumed to be strictly increasing and twice continuously differentiable. Then, \[ \rho _{X}\approx \dfrac{\sigma ^{2}}{2}\left( -\dfrac{u^{\prime \prime }(\omega)}{u^{\prime }(\omega)}\right). \]

Proof. By the definition of $\rho _{X}$, we have \[ \mathbb{E}[u(X)]=\int u(\omega+t)dF_{X}(t)=u(\omega-\rho _{X}). \] A Taylor expansion of $% u(\omega-\rho _{X})$ and $u(\omega+\varepsilon )$ for each realization of $% $ yields \[\begin{eqnarray*} u(\omega-\rho _{X}) &=&u(\omega)-\rho _{X}u^{\prime }(\omega)+\rho _{X}\delta (\omega,\rho )\\ &&\text{ where }\lim_{\rho \rightarrow 0}\delta (\omega,\rho )=0, \\ u(\omega+t) &=&u(\omega)+tu^{\prime }(\omega)+\frac{t^{2}}{2}u^{\prime \prime }(\omega)+t^{2}\delta ^{\prime }(\omega,t)\\ &&\text{ where }\lim_{t\rightarrow 0}\delta (\omega,t)=0. \end{eqnarray*}\] By integrating the above expression and equating it with the first expression, we obtain the desired approximation.

The previous result provides a local value for the risk premium (around a wealth level $\omega$), allowing us to isolate in the value of this premium an objective component (the variance of $$) and a subjective component (dependent on individual preferences).

The subjective component of the risk premium is called the absolute risk aversion coefficient and is denoted as $I_{A}(w)$ with \[\begin{equation*} I_{A}(\omega)=-\dfrac{u^{\prime \prime }(\omega)}{u^{\prime }(\omega)}. \end{equation*}\]

By definition, an individual will be more risk-averse than another if, for every random variable, their risk premium is greater than that of the second individual. This helps to understand why individuals, in identical situations, have different risk premiums due to their degree of risk aversion. Since 1738 and the work of Bernoulli, risk aversion has been associated with the concavity of the utility function. However, it was not until the work of and that the function $I_{A}$ measuring the strength of risk aversion as a function of the individual’s wealth appeared. The definition of $I_{A}$ is therefore local: it describes the individual’s willingness to hedge for small risks and can be seen as a risk premium per unit of variance. However, we establish below a certain duality between global and local preferences.

The following proposition (by Arrow and Pratt) allows for the comparison of attitudes toward risk between two individuals based on the characteristics of their utility functions.

Proposition 13.3 Let two decision-makers 1 and 2 satisfying the axioms of the expected utility model, characterized by the strictly increasing and twice continuously differentiable utility functions $u_{1}$ and $u_{2}$, respectively. The following statements are equivalent:

$\rho _{X}^{1}\geq \rho _{X}^{2};$
there exists $\varphi :$ ${\mathbb{R}}\rightarrow {\mathbb{R}}$, increasing and concave, such that $u_{1}=\varphi (u_{2});$
for all $ x,;I_{A}^{1}(x)I_{A}{2}(x).$

This result allows us to detect when a decision-maker exhibits more risk aversion than another. Indeed, the decision-maker whose utility function is $u_1$ is more risk-averse than the one whose utility function is $u_2$ since:

the degree of risk aversion is always higher with $u_1$ than with $u_2$;
the premium that the decision-maker with utility function $u_1$ is willing to pay to replace a risk with its mathematical expectation is always higher than what a decision-maker with utility function $u_2$ would be willing to pay;
the utility function $u_1$ is “more concave” than $u_2$.

To revisit the definition by , the notion of prudence expresses the agent’s desire to guard against risk, whereas the concept of risk aversion reflects their willingness to transfer it to another agent (such as an insurer). Therefore, for a risk-averse agent, prudent behavior can only arise when they cannot transfer the entire risk they bear. In this case, the way to manage the residual risk will differ depending on whether they exhibit prudent behavior or not.

In the framework of the expected utility model, the impact, in terms of utility, of a modification of the choice is measured by the expected marginal utility. A prudent agent is thus willing to pay a positive amount for the marginal utility of their decision to be certain rather than random.

In general, attitudes toward uncertainty about the outcome of a decision change are highlighted when considering the certain variable that has the same marginal utility as the risky lottery originally. Kimball defines, for a risk X and an initial wealth ω, a precautionary premium, by analogy with the Arrow-Pratt risk premium.

Definition 13.7 We call it a precautionary premium, denoted $\psi (X, \omega)$, the solution of the equation \[\begin{equation} \mathbb{E}[u^{\prime}(\omega+X)] = u^{\prime}(\omega+\mathbb{E}[X]-\psi(X, \omega)). \label{precautionary-premium} \end{equation}\]

Prudent behavior therefore corresponds to $\psi(X, \omega) > 0.$ Jensen’s inequality shows that

$\mathbb{E}[u^{\prime}(\omega+X)] \leq u^{\prime}(\omega+\mathbb{E}[X])$ if $u^{\prime}(\cdot)$ is concave,
$\mathbb{E}[u^{\prime}(\omega+X)] \geq u^{\prime}(\omega+\mathbb{E}[X])$ if $u^{\prime}(\cdot)$ is convex,
$\mathbb{E}[u^{\prime}(\omega+X)] = u^{\prime}(\omega+\mathbb{E}[X])$ if $u^{\prime}(\cdot)$ is linear, which corresponds to a quadratic utility function.

It is noted that the characterization of prudent behavior, in terms of properties of the utility function, depends on the attitude towards risk. For a risk-averse agent ($u^{\prime \prime}(\cdot) < 0$), we have:

$\psi(X, \omega) > 0$ if $u^{\prime}(\cdot)$ is convex,
$\psi(X, \omega) < 0$ if $u^{\prime}(\cdot)$ is concave,
$\psi(X, \omega) = 0$ if $u^{\prime}(\cdot)$ is linear, which corresponds to a quadratic utility function.

It follows that, within the framework of the expected utility model, the prudent behavior of a risk-averse agent is characterized by a convex marginal utility: $u^{\prime\prime\prime}(\cdot) > 0.$

Similar to how Arrow and Pratt deduced the absolute risk aversion coefficient from the risk premium, Kimball deduces the absolute and relative prudence coefficients from the precautionary premium. An expansion on both sides of equation ($\ref{precautionary-premium}$) yields an approximate but explicit value for the precautionary premium: \[\begin{equation*} \psi(X, \omega) \approx -\frac{1}{2}\mathbb{V}[X]\frac{u^{\prime\prime\prime}(\omega+\mathbb{E}[X])}{u^{\prime\prime}(\omega+\mathbb{E}[X])}. \end{equation*}\] We define $\eta_A$ (absolute prudence coefficient) from this expression as: \[\begin{equation*} \eta_A(\mathbb{E}[X], \omega) = -\frac{u^{\prime\prime\prime}(\omega+\mathbb{E}[X])}{u^{\prime\prime}(\omega+\mathbb{E}[X])}. \end{equation*}\]

13.4.2 Measures of Prudence and Aversion

Arrow and Pratt introduced the notions of absolute aversion and relative aversion in the mid-1960s as follows: \[ I_{A}(\omega) = -\frac{u^{\prime\prime}(\omega)}{u^{\prime}(\omega)} \text{ and } I_{R}(\omega) = -\frac{\omega u^{\prime\prime}(\omega)}{u^{\prime}(\omega)}. \] introduced the notions of absolute prudence and relative prudence, measured as: \[ \eta_{A}(\omega) = -\frac{u^{\prime\prime\prime}(\omega)}{u^{\prime\prime}(\omega)} \text{ and } \eta_{R}(\omega) = -\frac{\omega u^{\prime\prime\prime}(\omega)}{u^{\prime\prime}(\omega)}. \]

The absolute prudence coefficient allows for the comparison of the prudence level between two agents. It is noteworthy that the precautionary premium increases with the absolute prudence coefficient. Therefore, if an agent 1 has a positive and higher absolute prudence coefficient for every level of wealth compared to an agent 2, the corresponding precautionary premium for agent 1 will always be higher than that of agent 2, regardless of wealth or risk X.

In summary, schematically, it can be noted that some agents prefer:

certain situations over risky situations (notion of weak risk aversion),
situations with “low” risk over situations with “high” risk (notion of strong risk aversion),
situations with “low” uncertainty over situations with “high” uncertainty (notion of strong uncertainty aversion or prudence).

13.5 Insurance Supply and Demand

13.5.1 Normalization of Behaviors

A theory of insurance can be derived from the expected utility theory. As early as 1834, Théodore Barrois constructed a comprehensive theory of fire insurance based on the logarithmic utility function used by Bernoulli. However, it was only after the work of von Neumann and Morgenstern that utility theory was successfully applied to insurance contracts. In the early 1960s, Borch highlighted the relevance of this theory for actuaries in numerous works that had a significant influence on modern actuarial science.

The main contribution of this approach is to normalize the behaviors of economic agents. For instance, considering an economic agent with a utility function $u$ whose wealth is subject to a risk $X$, they will decide to purchase an insurance contract as long as this contract increases their average utility, i.e., as long as $u(\omega - \pi) \geq \mathbb{E}[u(\omega - X)]$, where $\omega$ is the agent’s wealth (which can be random and potentially correlated with $X$), and $\pi$ is the premium the insured must pay to obtain coverage from the insurer.

13.5.2 Insurance Demand

Models of insurance demand focus on the amount of coverage purchased by an individual to protect against a risk, considering different types of contracts and examining the impact of individual characteristics on insurance demand. For this purpose, we consider an individual with an initial wealth of $\omega$ facing a risk represented by a random variable $X \in [0, S]$ with $S \leq \omega$. Here, $S$ could represent the price of a car that might be damaged or the value of a house that could catch fire. The individual is assumed to be risk-averse, with a strictly increasing and concave utility function $u$. The insurance company is assumed to be risk-neutral and offers its products at a price determined by the market.

In its most general form, an insurance contract is characterized by a fixed premium (paid by the insured to the insurer) and a contingent benefit (paid by the insurer to the insured) based on the amount of damage suffered. We denote $\pi [I(\cdot)]$ as the premium corresponding to a benefit $I(X)$ (see Section 5.5 of Volume 1). In particular, if the insurer follows the principle of mathematical expectation (Definition 4.4.2), $\pi [I(\cdot)] = (1 + \lambda)\mathbb{E}[I(\cdot)]$ (as we will often assume later).

Furthermore, it is assumed that both over-insurance (where the benefit cannot exceed the loss amount) and negative insurance (where the insured cannot be required to pay anything to the insurer in the event of a loss) are prohibited, in accordance with most regulatory standards. Formally, this corresponds to $0 \leq I(x) \leq x$ for any $x \geq 0$.

13.5.3 The Mossin Model

We consider the case where either no loss occurs ($X=0$) or it completely destroys the insured property ($X=S$). Thus, \[ X=\left\{ \begin{array}{l} 0,\text{ with probability }1-q,\\ S,\text{ with probability }q. \end{array} \right. \] The utility function $u$ of the individual is assumed to be concave (the insured is risk-averse), increasing, and twice continuously differentiable. In the absence of insurance, the expected utility is \[ v_0=(1-q)u(\omega)+qu(\omega-S) \] where $\omega$ is the wealth of the insured (assumed to be known and constant). If the insured purchases a comprehensive insurance policy with a premium of $\pi$, the average utility becomes $v=u(\omega-\pi)$. Therefore, they will subscribe to this contract if $v>v_0$. This will be the case if $\pi$ does not exceed a certain amount $\pi_{\max}$ defined implicitly by the equation $u(\omega-\pi_{\max})=v_0$, beyond which the premium is considered too high. We can then obtain the following result.

::: {.proposition #DemandAss} The premium $\pi_{\max}$ satisfies 1. $\pi_{\max}>qS$; 2. $\pi_{\max}$ is an increasing function of $S$ and $q$; 3. if the risk aversion coefficient is decreasing, then $\pi_{\max}$ is a decreasing function of $\omega$. ::: ::: {.proof} The inequality in point (i) is easily deduced since \[ u(\omega-\pi_{\max})=v_0=\mathbb{E}[u(\omega-X)]\leq u(\omega-\mathbb{E}[X])=u(\omega-qS). \] Moving on to (ii), it is clear that $v_0$ is a decreasing function of $q$ since \[ v_0=q\underbrace{\big(u(\omega-S)-u(\omega)\big)}_{\leq 0\text{ because }u \text{ is increasing} }+u(\omega). \] Moreover, $v_0$ is also a decreasing function in $S$, again due to the growth of $u$. The result is then obtained from the implicit definition of $\pi_{\max}$. To establish (iii), start from the implicit equation giving $\pi_{\max}$ as a function of $\omega$, with $q$ and $S$ fixed, namely \[\begin{equation} \label{TourneBranle} u\big(\omega-\pi_{\max}(\omega)\big)=qu(\omega-S)+(1-q)u(\omega). \end{equation}\] Differentiating with respect to $\omega$, we obtain \[ u'(\omega-\pi_{\max}(\omega))\left(1-\frac{d}{d\omega}\pi_{\max}(\omega)\right)= qu'(\omega-S)+(1-q)u'(\omega) \] from which we deduce \[ \frac{d}{d\omega}\pi_{\max}(\omega)=1-\frac{qu'(\omega-S)+(1-q)u'(\omega)}{u'(\omega-\pi_{\max}(\omega))}. \] As a result, $d\pi_{\max}(\omega)/d\omega$ has the sign of \[\begin{eqnarray*} \psi(\pi_{\max})&=&u'(\omega-\pi_{\max})-qu'(\omega-S)-(1-q)u'(\omega). \end{eqnarray*}\] By extracting the value of $q$ from equation $\eqref{TourneBranle}$, we obtain \[\begin{eqnarray*} \psi(\pi_{\max})&=&u'(w-\pi_{\max})-\frac{u(w)-u(\omega-\pi_{\max})}{u(\omega)-u(\omega-S)}u'(\omega-S)\\ &&-\frac{u(\omega-\pi_{\max})-u(\omega-S)}{u(\omega)-u(\omega-S)}u'(w\omega). \end{eqnarray*}\] Note that $\psi(0)=\psi(S)=0$, so \[ \frac{d}{d\pi_{\max}}\psi=-u''(\omega-\pi_{\max})+u'(\omega-\pi_{\max})\frac{u'(\omega-S)-u'(\omega)}{u(\omega)-u(\omega-S)} \] vanishes at least once on the interval $]0,S[$. Since risk aversion is decreasing (i.e., $-u''/u'$ decreases), any root of $d\psi/d\pi_{\max}$ is a maximum. Consequently, $\psi\geq 0$ on $[0,S]$, from which we finally conclude that $d\pi_{\max}(\omega)/d\omega\leq 0$, as announced. :::

Commentary on These Results

tells us that insurance can only generate profits if individuals are risk-averse. (ii) indicates that the maximum acceptable premium is an increasing function of risk; i.e., if $q$ and/or $S$ increase, $\pi_{\max}$ will also increase. Finally, (iii) implies that if, as commonly accepted, risk aversion decreases with wealth, the maximum acceptable premium decreases with the insured’s wealth. But is full insurance the best possible coverage for an insured, or does it depend on the insured’s risk aversion?

Example 13.6 Suppose the insurer charges the insured a premium of $(1+\lambda)\mathbb{E}[X]$, where $\lambda$ represents the loading factor that transforms the pure premium into the commercial premium. The cost of insurance is therefore proportional to the size of the risk. The risk premium, on the other hand, is proportional to the variance of the risk. It varies as the square of the size of the risk. This linear increase in the cost of insurance compared to the quadratic increase in insurance profits with the size of the risk explains why only significant risks are insured. For example, only new cars are insured for physical damage. As the value of the vehicle decreases, their owners opt for liability coverage only.

Therefore, total insurance is not necessarily the best choice for the insured. It was Mossin who showed as early as 1968 that 100% insurance is never optimal when the amount demanded by the insurer exceeds the pure premium. The argument is the same: the cost of insurance is proportional to the coverage level, while its benefit is a quadratic function of it. Thus, going up to the last euro of coverage has zero marginal benefit, whereas its marginal cost is positive.

13.5.4 The General Model of Insurance Demand

We will now consider a slightly more general framework than the previous one, assuming that the amount of loss $X$ can take any value in the interval $[0, S]$. The commercial premium will include a loading factor proportional to the pure premium $\pi[I(X)] = (1 + \lambda) \mathbb{E}[I(X)]$ with $\lambda \geq 0$, where $\mathbb{E}[I(X)]$ is the pure premium. Agents are assumed to behave according to the expected utility model and are risk-averse: they have a utility function $u$, twice continuously differentiable, increasing, and concave.

Insurers offer two types of contracts: contracts, such that $I_\gamma(X) = \gamma X$, and contracts $d\in [0, S]$, characterized by the indemnity $I_{d}(X) = (X-d)_+$. These are the two cases that will be discussed in the following sections, inspired by .

13.5.5 Special Case of Proportional Insurance

The random wealth $W(\gamma, X)$ of an insured agent, insured for a proportion $\gamma$, is given by: \[\begin{equation*} W(\gamma, X) = \omega - (1-\gamma)X - (1+\lambda)\gamma \mathbb{E}[X]. \end{equation*}\] Thus, the optimal coverage is characterized by a proportion $\gamma^*$ that is a solution to the program:

\[\begin{equation} \max_{\gamma \in [0,1]}{\mathbb{E}[u(W(\gamma, X))]} \label{programme-assur-prop-optimale} \end{equation}\]

Proposition 13.4 The characteristics of the optimal proportional contract depend on the loading factor, namely:

In the absence of loading $(\lambda = 0)$, the optimal coverage is full insurance $(\gamma^* = 1);$
When the insurance premium is loaded $(\lambda > 0)$, the optimal coverage is partial insurance $(\gamma^* < 1)$.

Proof. The first and second-order conditions of the problem ($\ref{programme-assur-prop-optimale}$) can be written, if we do not consider the constraint $\gamma \in [0,1]$: \[\begin{align*} \frac{d\mathbb{E}[u(W(\gamma, X))]}{d\gamma} &= \mathbb{E}\left[(X-(1+\lambda)\mathbb{E}[X])u'(W(\gamma, X))\right] = 0, \\ \frac{d^2\mathbb{E}[u(W(\gamma, X))]}{d\gamma^2} &= \mathbb{E}\left[(X-(1+\lambda)\mathbb{E}[X])^2u''(W(\gamma, X))\right] < 0. \end{align*}\]

The second-order condition is satisfied thanks to the concavity of the utility function $u$.

There are two possible proofs for the result, one using the value of the first-order condition at $\gamma = 1$ and the other based on properties of covariance. The first one is presented here. We have:

\[\begin{equation*} \frac{d\mathbb{E}[u(W(\gamma, X))]}{d\gamma} = \mathbb{C}[u'(W(\gamma, X)), X] - \lambda \mathbb{E}[X]\mathbb{E}[u'(W(\gamma, X))]. \end{equation*}\]

\[\begin{eqnarray*} \left. \frac{d\mathbb{E}[u(W(\gamma, X))]}{d\gamma}\right|_{\gamma = 1} &=& \mathbb{C}[u'(cste), X] - \lambda \mathbb{E}[X]\mathbb{E}[u'(W(1, X))]\\ &=& -\lambda \mathbb{E}[X]\mathbb{E}[u'(W(1, X))] \end{eqnarray*}\]

So,

\[ \lambda = 0 \Rightarrow \left. \frac{d\mathbb{E}[u(W(\gamma, X))]}{d\gamma}\right|_{\gamma = 1} = 0 \Rightarrow \gamma^* = 1, \]

and

\[ \lambda > 0 \Rightarrow \left. \frac{d\mathbb{E}[u(W(\gamma, X))]}{d\gamma}\right|_{\gamma = 1} < 0 \Rightarrow \gamma^* < 1, \]

which proves the stated results.

13.5.5.1 Brief Commentary on These Results

If the insurer’s loading factor is positive, only risk-averse individuals insure themselves (those with a linear utility function are indifferent to risk and refuse to insure because they are asked to pay more than the pure premium). Furthermore, if the insurer decides to exclude a loading factor from its premiums (and therefore decides to charge insured individuals the pure premium), risk-averse insured individuals will opt for the most comprehensive coverage available. Finally, if the company includes a positive loading factor, rational insured individuals will forgo full coverage of their loss and opt for a proportional deductible.

13.5.5.2 Effect of Wealth $\omega$ on the Optimal Coverage $\gamma^*$

The following results examine the impact of variations in wealth and loading rate on insurance demand behavior. This is a standard study in microeconomics concerning the effect of wealth and price on the demand for a good. It helps determine the nature of insurance as a good.

Proposition 13.5 The impact of wealth on insurance demand primarily depends on changes in the agent’s risk attitude based on their wealth. Let $\lambda >0.$

$\dfrac{\partial \gamma^*}{\partial w} < 0$ if $u$ is a Decreasing Absolute Risk Aversion (DARA) function $(I_{A}'(w) < 0),$ meaning absolute risk aversion decreases.
$\dfrac{\partial \gamma^*}{\partial w} = 0$ if $u$ is a Constant Absolute Risk Aversion (CARA) function $(I_{A}'(w) = 0),$ meaning absolute risk aversion remains constant.
$\dfrac{\partial \gamma^*}{\partial w} > 0$ if $u$ is an Increasing Absolute Risk Aversion (IARA) function $(I_{A}'(w) > 0),$ meaning absolute risk aversion increases with wealth.

Proof. To obtain the result, we use the implicit function theorem:

\[\begin{equation*} \frac{\partial \gamma^*}{\partial w} = -\left(\frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma \partial w}\right)\left(\frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma^2}\right)^{-1}, \end{equation*}\]

and we study the sign of $\partial^2\mathbb{E}[u(W(\gamma, X))]/\partial \gamma \partial w$. To do this, let’s write:

\[\begin{equation*} \left.\frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma \partial w}\right|_{\gamma^*} = \int_{0}^{S}u''(W(\gamma^*, x))(x-(1+\lambda)\mathbb{E}[X])dF_X(x). \end{equation*}\]

Let $x_0 = (1+\lambda)\mathbb{E}[X]$ and $y_0 = w - x_0$, and multiply the last expression by $-\dfrac{u'(W(\gamma^*, x))}{u'(W(\gamma^*, x))}$. We then obtain:

\[\begin{eqnarray*} &&\left.\frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma \partial w}\right|_{\gamma^*} \\ &=& -\int_{0}^{x_0}I_A(W(\gamma^*, x))u'(W(\gamma^*, x))(x-x_0)dF_X(x) \\ &&-\int_{x_0}^{S}I_A(W(\gamma^*, x))u'(W(\gamma^*, x))(x-x_0)dF_X(x) \\ &<& -I_A(y_0) \int_{0}^{S}u'(W(\gamma^*, x))(x-(1+\lambda)\mathbb{E}[X])dF_X(x) = 0, \end{eqnarray*}\]

where the inequality results from the decreasing property of $I_A(x)$ in the case of a DARA utility function. This proves (i). The reasoning leading to (ii) and (iii) is analogous.

However, it is worth noting that the CARA, DARA, and IARA utility functions do not form a partition of the set of utility functions. Therefore, the conditions in the proposition are sufficient but not necessary.

13.5.5.3 Effect of Loading Rate $\lambda$ on the Optimal Coverage $\gamma^*$

The impact of the loading rate on insurance demand is generally indeterminate. Indeed,

\[ \dfrac{\partial \gamma^*}{\partial \lambda} = -\left(\frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma \partial \lambda}\right)\left(\frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma^2}\right)^{-1}, \]

and thus,

\[ \text{sign}\left(\dfrac{\partial \gamma^*}{\partial \lambda}\right) = \text{sign}\left(\frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma \partial \lambda}\right). \]

\[\begin{equation*} \frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma \partial \lambda} = -\mathbb{E}[X]\mathbb{E}[u'(W(\gamma, X))] - \gamma \mathbb{E}[X]\frac{\partial^2\mathbb{E}[u(W(\gamma, X))]}{\partial \gamma \partial w}, \end{equation*}\]

an increase in the loading rate results in two opposing effects: the substitution effect and the income effect. When the loading rate increases, the marginal cost of insurance increases, and individuals substitute risk retention for insurance, leading to a decrease in insurance demand. Additionally, increasing coverage leads to a real impoverishment of the individual since their wealth decreases, whether or not there is a claim. Thus, if the individual has a decreasing absolute risk aversion, the reduction in wealth will lead to an increase in risk aversion, and they will desire increased coverage. If absolute risk aversion decreases rapidly, the income effect may outweigh the substitution effect, and an increase in the loading rate may lead to an increase in coverage. In this case, insurance is a Giffen good (or inferior good).

Proposition $\ref{impactwgamma}$ directly implies the following result.

Proposition 13.6 Let $\lambda >0$ and $0 < \gamma^* < 1$. If an individual’s preferences are either CARA or IARA, insurance demand decreases with an increase in the loading rate. If preferences are DARA, insurance demand may increase with an increase in the loading rate.

13.5.5.4 Effect of Risk Aversion on the Optimal Coverage $\gamma^*$

We now investigate the impact of changes in risk and risk aversion on insurance demand. The following result shows that an increase in risk aversion always leads, as intuitively expected, to an increase in coverage.

Proposition 13.7 Suppose $\lambda >0$ and $0 < \gamma^* < 1$. Consider two individuals, 1 and 2, who differ only in their utility functions. Let $u_1$ be the utility function of individual 1, and $u_2$ be the utility function of individual 2, with $u_2 = \varphi(u_1)$, where $\varphi$ is increasing and concave. If $\gamma_1^*$ and $\gamma_2^*$ are the optimal coverage amounts for individuals 1 and 2, respectively, then $\gamma_2^* > \gamma_1^*$.

Proof. Both $\mathbb{E}[u_i(W(\gamma, X))]$ are concave in $\gamma$ for $i=1,2$. Thus, to prove the result, it suffices to show:

\[\begin{equation*} \left.\frac{d\mathbb{E}[u_2(W(\gamma, X))]}{d\gamma}\right|_{\gamma = \gamma_1^*} > 0. \end{equation*}\]

Using the concavity of the function $\varphi$, we can write:

\[\begin{eqnarray*} &&\left.\frac{d\mathbb{E}[u_2(W(\gamma, X))]}{d\gamma}\right|_{\gamma = \gamma_1^*} \\ &=& \left.\frac{d\mathbb{E}[\varphi\{u_1(W(\gamma, X))\}]}{d\gamma}\right|_{\gamma = \gamma_1^*} \\ &=& \int_{0}^{S}\varphi'\{u_1(W(\gamma_1^*, X))\}u_1'(W(\gamma_1^*, X))(x-(1+\lambda)\mathbb{E}[X])dF_X(x) \\ &>& \varphi'\{u_1(y_0)\}\int_{0}^{S}u_1'(W(\gamma_1^*, X))(x-(1+\lambda)\mathbb{E}[X])dF_X(x) = 0. \end{eqnarray*}\]

From the concavity of $\mathbb{E}[u_2(W(\gamma, X))]$ in $\gamma$, it follows that $\gamma_2^* > \gamma_1^*$.

13.5.6 Special Case of Insurance with Mandatory Deductible

In the case of insurance with a mandatory deductible, the insured is only compensated for losses above a certain amount specified in the contract (see Section 5.5.4). Their random wealth $W(d, X)$ in the case of purchasing an insurance contract with a mandatory deductible of amount $d$ can be expressed as:

\[\begin{equation*} W(d, X) = \omega - X + (X - d)_+ - (1 + \lambda)\mathbb{E}[(X - d)_+], \end{equation*}\]

assuming that the premium is determined with a proportional loading.

Recall that total coverage corresponds to $d = 0$ in this case, where all losses are covered, and the absence of insurance corresponds to the case $d = S$, where no losses are covered. The program for determining the optimal amount of mandatory deductible is formulated as:

\[\begin{equation} \max\limits_{d \in [0, S]}\{\mathbb{E}[u(W(d, X))]\} \label{optimal deductible} \end{equation}\]

We obtain a similar result as for proportional insurance.

Proposition 13.8 The characteristics of the optimal contract with a mandatory deductible depend on the loading factor, namely:

In the absence of loading ($\lambda = 0$), the optimal coverage is full insurance ($d^{\ast} = 0$);
When the insurance premium is loaded ($\lambda > 0$), the optimal coverage is partial insurance ($d^{\ast} > 0$).

Proof. First, note that integration by parts gives: \[\begin{equation*} \pi (d, X) = (1+\lambda )\left( (S-d) - \int\limits_{d}^{S} F_X(x)dx \right). \end{equation*}\] Furthermore, $\dfrac{\partial \pi (d, X)}{\partial d} = -(1+\lambda)\overline{F}_X(d).$ Additionally, \[\begin{equation*} \mathbb{E}[u\left( W(d, X)\right)] = \int_{0}^{d} u(\omega-x-\beta(d, X))dF_X(x) + \overline{F}_X(d) u(\omega-d-\pi(d, X)). \end{equation*}\] We assume that the loading factor is sufficiently low so that zero insurance is not optimal.

The first-order condition of the program ($\ref{optimal deductible}$) can be written, after some transformations, as: \[\begin{eqnarray*} &&\frac{d\mathbb{E}[u\left( W(d, X)\right)]}{dd} \\ &=& -u^{\prime}(\omega-d-\beta(d, X))\overline{F}_X(d) - \dfrac{\partial \pi(d, X)}{\partial d}\mathbb{E}[u^{\prime}\left( W(d, X)\right)] \\ &=& \overline{F}_X(d) \left\{ (1+\lambda)\mathbb{E}[u^{\prime}\left( W(d, X)\right)] - u^{\prime}(\omega-d-\beta(d, X))\right\} = 0. \end{eqnarray*}\]

The second-order condition is satisfied due to the concavity of the utility function.

The proof of the proposition relies on the fact that for any $x \in [0, S]$ and for any $d$, we have: \[\begin{equation*} W(d, x) \geq \omega-d-\pi(d, X). \end{equation*}\]

The following cases can then occur: 1. If $d=0$, then $W(d, x) = \omega-d-\pi(d, X)$ for all $x$. 2. If $d>0$, then there exists at least one $\widehat{x}$ such that $W(d, \widehat{x}) > \omega-d-\pi(d, X)$, and therefore $\mathbb{E}[u^{\prime}\left[ W(d, X)\right]] < u^{\prime}(\omega-d-\pi(d, X))$.

It follows directly that: - If $\lambda = 0$, then the first-order condition can only be satisfied if $d=0$. - If $\lambda > 0$, then at $d=0$, we have \[ (1+\lambda)\mathbb{E}[u^{\prime}\left( W(d, X)\right)] > u^{\prime}(\omega-d-\beta(d, X)), \] to obtain equality, $d$ must be increasing.

This completes the proof.

13.6 Information Asymmetry and Adverse Selection

13.6.1 Incomplete Information

The nature of risk, agent preferences, insurer management costs, and market rules all determine the characteristics of exchanges, their efficiency, and the distribution of the generated surplus. However, there is no guarantee that the agents’ own risks or their preferences are known as well to insurers as they are to the insured.

The quantity of data an insurer possesses about the claims experience of a large number of individuals can create (see Section 3.8.7 for a precise definition of this concept): certain statistical tools allow insurers, using this data, to create homogeneous groups of individuals and reliably estimate the risk of members within each of these groups. An insurer may know better than anyone else the probability of having a car accident in a year (see Chapter 9 on a priori pricing). Conversely, the insured may also have information that the insurer does not possess, constituting information asymmetry in the other direction. In health insurance, a lot of information is known by the insured alone.

The presence of information asymmetry (of which all agents are aware) alters their strategic behaviors and therefore has an impact on the characteristics of the risk transfer mechanism.

Remark. Most insurance professionals share the impression that insured individuals are far from the rationality underlying the Bernoulli principle-based approach. Nevertheless, this does not question the interest of insurance microeconomics. Indeed, as have pointed out, it is extremely difficult to distinguish incomplete information from a lack of rationality. In general, insured individuals have a very vague perception of the risks they face (they have immense difficulty evaluating the probability of experiencing an accident, and even more so the extent of the resulting damage). Moreover, the insurance products and formulas available on the market are sometimes highly varied, and insured individuals often have only partial information about insurers’ offerings (which justifies the role of brokers, for example). The complexity of these products and the general public’s lack of understanding of basic probability calculations clearly contribute to the opaqueness of the market. Even if the consumer is rational, they will probably settle for a solution that provides them with a reasonable level of utility, even if they know that better options exist on the market.

13.6.2 Adverse Selection, Moral Hazard, and Signals

Models that study equilibrium exchanges in the presence of asymmetric information can be grouped into several categories. The first distinction that can be made is whether it is the insurer or the insured who possesses the information. Another element that allows for a finer classification is the type of information involved. This leads to the following major groups, each using specific resolution methods:

When uninformed participants take the initiative in proposing a transaction, and the information concerns the unchangeable characteristics of agents, we refer to models. Typically, this occurs when insured individuals know their risk, and it is up to them to decide whether or not to purchase an insurance contract.
When uninformed participants take the initiative, but the information pertains to the actions of agents, we refer to . Typically, an individual insuring their car in a risky location creates a situation of moral hazard.
When the initiative belongs to the best-informed agents, we refer to . Typically, insurers can offer contracts with or without deductibles to specific clients.

The importance of adverse selection in insurance, although formalized relatively late, has been a known issue for a long time. As early as 1895, the actuary Miles Dawson noted that, “ [and hence the least risky] ”

Similarly, moral hazard has been recognized by actuaries for a long time as well. As cited by , a practical insurance guide from the 1840s noted that a significant portion of claims was due to criminal or “self-interested inattention,” no longer entirely due to chance. These deemed immoral behaviors led insurers to use the term “moral hazard” as early as 1860.

13.6.3 The Rothschild & Stiglitz Equilibrium Model

In this chapter, we focus on cases where information asymmetry pertains to the objective characteristics of agents and not their actions.

The analysis of the competitive insurance market in the presence of insured individuals’ private information about their objective risk has developed from the pioneering article by . The authors assume that there are two risk groups in the population that the insurer cannot distinguish between, and they demonstrate that, at equilibrium (if it exists), the insurer offers a different contract for each risk group. The identification of equilibrium non-existence situations in this context has directed research toward finding mechanisms to address this inefficiency. Several authors have proposed solutions, more or less credible, to address the absence of equilibrium. Another direction in which research on the subject has ventured is the study of repeated contracts, both with full commitment and with renegotiation possibilities: having or not having an accident during a given period allows the insurer to revise its beliefs about an agent’s risk and modify the contract terms when renegotiation is possible.

Research has thus focused for about fifteen years on private information concerning an agent’s risk, rather than their preferences. This can be explained by a hierarchy, from the insurer’s perspective, of agent characteristics. The probability of loss is crucial for the insurer because it determines what an insurance contract costs: for a given number of identical individuals, it determines the number of those who will have an accident and thus the indemnities the insurer will have to pay. A significant error in the probability of loss for a large number of insured individuals can have dramatic consequences (since the premium is calculated on a wrong basis, the insurer may struggle to pay all necessary indemnities).

Taking into account agents’ heterogeneity in terms of preferences is also very important because these preferences determine agent choices. Its role becomes especially apparent when the insurer offers different contracts to different risk groups to reveal objective risks. In such cases, if agents’ preferences differ without the insurer taking them into account, agent choices may differ from those anticipated by the insurer. The study of cases where agents have private information about their preferences poses notable technical challenges and has only been recently studied by , in the case where the insurer is in a monopoly situation.

13.6.4 Study of Adverse Selection Mechanisms

In this first part, we will consider the case of an insurer monopoly. The more complex case with competition among insurers will be discussed at the end of the section.

13.6.4.1 Asymmetric Information Concerning the Probability of Loss

In this model, we consider a population of individuals within the framework of a utility expectation model. The insurer knows that within this population, two levels of risk are possible and is aware of these two levels and their proportions in the population. However, when an individual presents themselves, the insurer is unable to identify them as having either of the two risk levels. The individual, on the other hand, knows their risk perfectly (in the literature on the subject, the term “type” of the agent is often used rather than “risk”). Special cases include mixture Poisson models (Section 3.7.2).

The study of insurance contracts in the presence of information asymmetry when the insurer is in a monopoly situation is based on Principal-Agent models, where the insurer takes on the role of the Principal, who initiates the proposal of one or multiple contracts, and the insured is the Agent who possesses private information. The insurer can offer a single contract or a menu of contracts.

The two types of agents in the economy are denoted as $H$ (high risk) and $B$ (low risk). They both face the risk of a loss of a unique amount $S$, have an initial wealth of $\omega$, and their attitude toward risk is represented by the same utility function $u$, assumed to be increasing and concave. Their loss probabilities are denoted as $p_{H}$ and $p_{B}$, respectively, with $p_{H} > p_{B}$, and the group of type $H$ agents represents a proportion $\lambda$ of the population.

Before determining the characteristics of optimal contracts, we demonstrate a property of agent preferences that allows for the offering of separating contracts based on the fact that one of the two types of agents is willing to pay more than the other for an increase in their coverage: the indifference curves of agents $H$ and $B$ intersect only once (a condition known as single crossing).

When an insurance company is unable to recognize an individual’s risk when they come to purchase a contract, it seeks to determine pairs of contracts (premium-payout) that maximize its profit while taking into account that an agent only buys a contract if they prefer it to non-insurance (participation constraint) and only chooses the contract intended for them if they prefer it to the contract offered to the other type of agent (revelation constraint). The results from can be summarized as follows.

Proposition 13.9

The optimal contract for high risks is full insurance.
If low risks purchase a non-zero insurance contract, their utility is the same as if they did not purchase insurance.
High risks are indifferent between the contract intended for them and the contract intended for low risks.
The contracts offered to high risks and low risks are always different.
There exists a finite proportion of high risks such that above this critical proportion, no contracts are offered to low risks.

13.6.4.2 Asymmetric Information Concerning Preferences

The study of asymmetric information about individuals’ preferences has attracted less attention because the non-accounting of this asymmetry has less severe consequences for the insurer’s profits than the asymmetry in risk characteristics. Indeed, the informational advantage of agents regarding their risk can lead to losses for the company, whereas when it comes to preferences, it is merely a missed opportunity, and that only for a company in a monopoly situation.

The results presented below even show that, in certain situations, it is optimal for the company not to attempt to reveal information about preferences and to offer a single contract. The model is more general than that of Stiglitz, with loss amounts being continuous variables.

The population consists of two types of risk-averse individuals who have the same random wealth $W$ (a continuous random variable) but differ in the intensity of their risk aversion. Type $H$ individuals are more risk-averse than type $B$ individuals (in the Arrow-Pratt sense). Type $H$ individuals have a utility function $v$ with $v^{\prime}>0$ and $v^{\prime\prime}<0$. They constitute a proportion $\lambda$ of the population. Type $B$ individuals have a utility function $u$ with $u^{\prime}>0$ and $u^{\prime\prime}<0$.

We can then obtain the following result, due to .

Proposition 13.10

In the absence of asymmetric information, the insurer maximizes its profit by offering both types of individuals their certain equivalents of non-insurance, i.e., $u^{-1}(\mathbb{E}[u(W)])$ for type $B$ individuals and $v^{-1}(\mathbb{E}[v(W)])$ for type $H$ individuals, where $W$ denotes the (random) wealth of individuals.
With asymmetric information, if $u(x)/v(x) \rightarrow 0$ as $x \rightarrow -\infty$, it is possible for the insurer to offer contracts that allow it to approximately achieve the first-best profit (without asymmetric information).
With asymmetric information, the proposed payouts are bounded, and the separating contracts that maximize profit are as follows:

Type $H$ individuals receive full coverage;
Type $B$ individuals receive partial coverage and are indifferent between their contract and non-insurance;
There exists $\lambda^*$ such that if $\lambda < \lambda^*$, only one complete insurance contract is offered: all agents receive $u^{-1}(\mathbb{E}[u(W)])$.

13.6.4.3 Adverse Selection and Competition

Once again, were the first to study competitive equilibrium in the insurance market in the presence of asymmetric information. They adopt the concept of Nash equilibrium, which they adapt to the insurance market. The competitive process is represented by a two-stage game: in the first stage, uninformed agents simultaneously offer one or more contracts, considering the offers of their competitors as given (each company’s strategy is a best response to the strategies of rivals). In the second stage, informed agents choose from these offered contracts. The definition of equilibrium in the insurance market that follows from this type of firm behavior is as follows.

Definition 13.8 (Nash-Rothschild-Stiglitz Equilibrium) An equilibrium in the insurance market is a set of contracts such that:

each contract is non-deficient;
there is no contract outside the set of equilibrium contracts that, if offered together with the equilibrium contracts, would be profitable.

Here, we assume that companies can impose , meaning they have the power to limit consumers’ purchases to a single contract. The model’s assumptions are identical to those of the previous section. The first-order conditions of this program naturally lead to the equality of the marginal rates of substitution of the insurer and the insured for each type of risk. Thus, equilibrium contracts in perfect information correspond to complete insurance contracts priced at the pure premium of each type.

In about the probabilities of loss, the high-risk individual, not recognized as such, will choose the contract intended for the low-risk individual that offers the same level of coverage for a lower premium. This leads to losses for the company. Therefore, in determining equilibrium contracts, we must now take into account the unobservability of individuals’ types by the insurer.

13.7 Coverage of Multiple Risks

13.7.1 Context

Chapter 8 of Volume 1 dealt in detail with the modeling of multiple risks. Here, we will explore how insurers behave in the presence of several risks, whether insurable or not.

Example 13.7 Let’s consider two risks that can be affected by a common first. For example, an insured wishing to cover their home and automobile may see a fire at the residence destroy both properties if the car was parked in the garage. In life insurance for couples, spouses may die simultaneously (in a car accident, for example). Thus, some risks are naturally . However, it is possible for two risks to be . Consider the case of automobile insurance, where one risk is excluded by the insurance policy (and therefore will be the responsibility of the insured), for example, the risk of vandalism. The fact that the insured still has to deal with this risk means that for them, the real risk is divided into two, one insurable and the other not. The sum of the two being bounded by the value of the insured property (the vehicle’s value), the risks are then . In this example, we actually have exclusive risks (see Section 8.2.3).

Recall that in the case of managing multiple risks, a number of properties are desirable, and they impose certain constraints. For example, if $X$ and $Y$ are two risks such that, if $u$ denotes the utility function of an agent, $\mathbb{E}[u(X)]\leq \mathbb{E}[u(Y)]$, it may seem natural that if $Z$ is independent of $X$ and $Y$, then \[ \mathbb{E}[u(X+Z)]\leq \mathbb{E}[u(Y+Z)]. \] This property, although natural, is by no means universal, as shown by the following result.

Proposition 13.11 The following properties are equivalent:

If $Y$ is preferred to $X$ in the sense that $\mathbb{E}[u(X)]\leq \mathbb{E}[u(Y)]$, then $\mathbb{E}[u(X+Z)]\leq \mathbb{E}[u(Y+Z)]$ for all $Z$ independent of $X$ and $Y$.
The utility function $u$ is CARA.

13.7.2 Insurable Risk and Non-Insurable Risk

showed that when insurable and non-insurable risks are independent, the insured chooses full coverage if the premium charged by the company does not include a loading, and partial coverage otherwise. In the case of positively dependent risks and if the premium does not include a loading, it is optimal to overinsure (if possible) the insurable risk so that indirect coverage against the non-insurable risk is obtained. Thus, the total risk is better controlled. If the insurance premium includes a loading, additional assumptions should be added. In the case of negative dependence, it is possible for the risks to cover each other, and individuals may then insure less (this is called self-insurance).

added additional information to the understanding of insurance mechanisms (considering the general case of continuous variables without seeking the optimal contract form). If prudence is positive and decreases with wealth, the presence of an independent or positively correlated non-insurable risk with the insurable risk will tend to increase the optimal coverage amounts for the latter.

More formally, consider an agent with initial wealth $\omega$, subject to two risks $X$ (insurable) and $Y$ (non-insurable) assumed to be independent. Let $I(X)$ denote the indemnity associated with the insurance contract covering $X$. The agent seeks to solve \[ \max\{\mathbb{E}\left[u(\omega-(1+\lambda)\mathbb{E}[I(X)]-(X+Y)+I(X))\right]\}. \] Also, if we denote $v(x)=\mathbb{E}[u(x-Y)]$, it can be clearly seen that $v$ is strictly concave and increasing. Due to the independence between risks $X$ and $Y$, it can be equivalently stated that the agent seeks to solve \[ \max\left\{\mathbb{E}\left[v(\omega-(1+\lambda)\mathbb{E}[I(X)]-X+I(X))\right]\right\}. \] This program corresponds to the one discussed in Section $\ref{offre-demande-section}$, and thus the results obtained previously are valid, particularly regarding the optimal form of contracts.

Proposition 13.12 In the presence of a non-insurable risk independent of the insurable risk, the optimal contract is one with mandatory deductible. In the special case where $\lambda=0$ (no loading), the optimal coverage is full insurance.

However, it may be legitimate to question whether the coverage is then greater than in the absence of non-insurable risk. To answer this question, it is necessary to add a number of assumptions about the utility function $u$.

Proposition 13.13 If $u$ is four times differentiable, DARA, and satisfies the DAP assumption (i.e., the absolute prudence coefficient $\eta_A$ is decreasing), then the presence of a non-insurable risk independent of the insurable risk will result in greater coverage.

However, as noted in the introduction, assuming that the non-insurable residual risk (also referred to as background risk) is independent of the insurable risk is a very strong assumption. As detailed in Chapter 8, characterizing the type of dependence can be relatively complex. limited their study to cases where the risks were Gaussian, and dependence was characterized by correlation. considered the case of third-order stochastic dominance. Finally, considered the case of concordance order (introduced in Section 8.4.1 of Volume 1), with strong assumptions about preferences.

13.7.3 Presence of Multiple Insurable Risks

In this case, based on financial models, one might think that if multiplicity allows for diversification, the demand for insurance could be less significant.

Consider again two risks, $X$ and $Y$, which we assume to be independent, both insurable but with distinct and separate coverages. The (random) wealth of the insured agent is then given by \[ W=\omega-X+I_X(X)-(1+\lambda_X)\mathbb{E}[I_X(X)]-Y+I_Y(Y)-(1+\lambda_Y)\mathbb{E}[I_Y(Y)]. \]

Recall that if the contract associated with risk $X$ includes a mandatory deductible, the premium is then written as \[ \pi(d_X)=(1+\lambda_X)\int_{d_X}^\infty(x-d_X)dF_X(x) \] which satisfies $\pi'(d_X)=-(1+\lambda_X)\overline{F}_X(d_X)<0$. Let $\gamma=\pi^{-1}$ be the inverse of the premium. Then \[ \gamma_X'(p)=\frac{-1}{(1+\lambda_X)\overline{F}_X(\gamma_X(p))}. \] The insured must then solve the following program: \[ \max\Big\{\mathbb{E}\Big[u\Big(\omega-(1+\lambda_X)\mathbb{E}[I_X(X)]-(1+\lambda_Y)\mathbb{E}[I_Y(Y)] \] \[ -X-Y+I_X(X)+I_Y(Y)\Big)\Big]\Big\}. \] ::: {.proof} The program that the insured must solve is of the form \[\begin{eqnarray*} &&\max \{\mathbb{E[}u(\omega -\left( 1+\lambda _{X}\right) \mathbb{E}\left[ \left( X-d_{X}\right) _{+}\right] -\left( 1+\lambda _{Y}\right) \mathbb{E}% \left[ \left( X-d_{Y}\right) _{+}\right] \\ &&\quad \quad -X-Y+\left( X-d_{X}\right) _{+}+\left( Y-d_{Y}\right) _{+})]\} \end{eqnarray*}\] which has a solution because the function%

\[\begin{eqnarray*} \left( d_{X},d_{Y}\right) &\mapsto &\mathbb{E}\left[ u(\omega -\left( 1+\lambda _{X}\right) \mathbb{E}\left[ \left( X-d_{X}\right) _{+}\right] -\left( 1+\lambda _{Y}\right) \mathbb{E}[\left( X-d_{Y}\right) _{+}\right] \\ &&-X-Y+\left( X-d_{X}\right) _{+}+\left( Y-d_{Y}\right) _{+})] \end{eqnarray*}\]% is continuous, and uniqueness is ensured by strict concavity. To show that the deductible contract is optimal, note that% \[\begin{eqnarray*} \Delta U &=&\mathbb{E[}u(\omega -\left( 1+\lambda _{X}\right) \mathbb{E}% \left[ \left( X-d_{X}\right) _{+}\right] -\left( 1+\lambda _{Y}\right) \mathbb{E}\left[ \left( X-d_{Y}\right) _{+}\right] \\ &&-X-Y+\left( X-d_{X}\right) _{+}+\left( Y-d_{Y}\right) _{+})] \\ &&-\mathbb{E[}u(\omega -\left( 1+\lambda _{X}\right) \mathbb{E}\left[ I_{X}\left( X\right) \right] -\left( 1+\lambda _{Y}\right) \mathbb{E}\left[ I_{Y}\left( Y\right) \right] \\ &&-X-Y+I_{X}\left( X\right) +I_{Y}\left( Y\right) )]. \end{eqnarray*}\]

Let’s then demonstrate that all other contracts (characterized by the indemnity functions $I_X$ and $I_Y$ respectively), such that $\mathbb{E}[I_X(X)]=\mathbb{E}[(X-d_X)_+]$, and $\mathbb{E}[I_Y(Y)]=\mathbb{E}[(Y-d_Y)_+]$, are dominated by the two proposed contracts. Some calculations show that \[ \mathbb{E}\Big[ u^{\prime }\Big( \omega -\pi _{X}\left( X\right) -\pi _{Y}\left( Y\right) -X-Y+I_{X}\left( X\right) + I_{Y}\left( Y\right) \Big) \left( X-d_{X}\right) _{+}-I_{X}\left( X\right) \Big] =0, \] and similarly, that \[ \mathbb{E}\Big[ u^{\prime }\Big( \omega -\pi _{X}\left( X\right) -\pi _{Y}\left( Y\right) -X-Y+I_{X}\left( X\right) + I_{Y}\left( Y\right) \Big) \left( Y-d_{Y}\right) _{+}-I_{Y}\left( Y\right) \Big] =0, \] using the independence between risks $X$ and $Y$. Also, $\Delta U$, which is necessarily greater than the sum of these two expressions, is always positive, with equality achieved for $I_{X}\left( x\right) =\left( x-d_{X}\right) _{+}$ and $I_{Y}\left( y\right) =\left( y-d_{Y}\right) _{+}$. :::

Example 13.8 In the case where the agent has a CARA utility function, it is easy to see that the optimal policies are the same as when the agent insures a single risk, $X+Y$.

Exercise 13.1 Consider an economic agent following the expected utility model, motivated by profit, and facing a risk $X$. An insurer is willing to cover this risk in exchange for a premium payment.

Show that there exists a limit $\pi_{\max}$, depending on the wealth $w$ of the insured, the risk $X threatening their wealth, and the insured’s preferences reflected by their utility function $u$, beyond which the insured will refuse to purchase the policy.
Show that $\mathbb{E}[X]\leq \pi_{\max}$ if the insured is risk-averse (i.e., if $u$ is concave).

Exercise 13.2 Consider the risk \[ X=\left\{ \begin{array}{l} m-h,\text{ with probability }\frac{1}{2},\\ m+h,\text{ with probability }\frac{1}{2}. \end{array} \right. \] Show that all risk-averse decision-makers agree that the risk worsens as its variance increases.

Exercise 13.3 Consider a decision-maker whose utility function is \[ u(x)=\frac{1-\exp(-cx)}{c},\hspace{2mm}c>0. \] 1. Is this decision-maker guided by profit? 2. Does this decision-maker exhibit risk aversion? 3. Show that the maximum amount this decision-maker would be willing to pay to insure the risk \[ X=\left\{ \begin{array}{l} 0,\text{ with probability }p,\\ s,\text{ with probability }1-p. \end{array} \right. \] is $\pi_{\max}=\frac{1}{c}\ln(p+(1-p)\exp(cs))$. \end{enumerate}