Estimating an optimal bonus-malus contract with asymmetric information.

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Estimating an Optimal Bonus-Malus Contract with

Asymmetric Information

Sonja Vogel 6064299

Msc Thesis Actuarial Science & Mathematical Finance University of Amsterdam

Supervisor: Dr. M.A.L. Koster Second Supervisor: Dr. T.J. Boonen

April 30, 2014

Abstract

This thesis provides a new perspective on bonus-malus contracts, by describing a signaling game between an individual and a car insurance firm. The insurer offers a bonus-malus contract which is characterized by a base premium, the percentage of bonus or malus in the different classes and the discount a policyholder receives when he takes a deductible. The individual can decide to take an insurance and whether or not to take a deductible. The insurer sets the parameters of the bonus-malus contract such that it maximizes its expected profit. The goal of this thesis is to find equilibria in this game. The main question is: What are the conditions for a separating equilibrium in which good drivers opt for a deductible and bad drivers do not? We find that a separating equilibrium provides higher expected profits for the insurer than a pooling equilibrium, as expected.

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1 Introduction

Bonus-malus systems are used to reward (bonus) or penalize (malus) the holder of the contract. Subject to car insurance this involves paying a lower (bonus) or higher (malus) premium as opposed to the base premium. In the Netherlands, car insurance bonus-malus (BM) systems generally have 15 to 20 BM classes. The higher the number of the class, the higher the discount the policyholder receives on the base premium. When policyholders do not claim any damage, they move to a higher class and therefore pay a lower premium. If they do make a claim, they are penalized by descending to a lower class and thus paying a higher premium. Through this method, the insurer tries to differentiate the good drivers from the bad drivers, which will in turn help to manage the risks. This thesis examines the bonus-malus system from an entirely new perspective.

Considerable research has been done in the field of bonus-malus schemes. Dellaert et al. (1990) derived optimal critical claim sizes for the policyholder when there is no deductible, considering multiple horizons, while Norman and Shearn (1980) obtained a claim or no-claim decision rule for car drivers. Kliger and Levikson (2002) found an optimal decision for the insurer in a situation where the policyholder first pays a base premium π and receives a discount d after X claim free years. The policyholder is always insured and its only decision is to set a threshold damage above which he will always claim. This is done by setting equal the EPVs (Expected Present Value) of both claiming and not claiming with the assumption of a constant discount value. In this thesis, a similar threshold damage is derived, which will be of use when deriving the payoffs in the signaling game.

Gomez et al. (2002) performed a Bayesian analysis on the prior belief of the insurer. They made the information set a convex combination of a fixed prior and a set of alternative priors. By analyzing the range of the premiums within a class, they showed that the choice of this prior can crucially affect the relative premiums in the different classes. Moreover, they noted that the range of variation of the relative premium is large, so that it is hard to set the right premium.

Majeske (2007) investigated the relation between policyholder behavior and the type of bonus-malus contract in the Turkish system. He found that there is no single rule of thumb which describes the optimal behavior of all different damage probabilities of the policyholders. He also stated that some policyholders might act sub-optimally due to a budget constraint.

All these studies look at optimal behavior for either the insured or the insurer, but never for both. This thesis is the first to so by investigating optimality for both agents at the same time. Obviously, a conflict of interest exists: while the insured wishes to pay as little for its insurance as possible, the insurer wants to maximize its profit. This conflict of interest can be translated into strategic actions. Moreover, the payoffs depend on the actions of the other player. Hence, a game theoretical approach is suitable for this situation. Furthermore, there is information asymmetry, as the individual has information on its driving skills, which the insurer has not. Therefore, the

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interaction between the insurer and the individual is modeled as a signaling game. Specifically, it is modeled as a signaling game in which the policyholder sends a signal about its driving skills by means of choosing a deductible or not.

In a signaling game two players participate; a sender and a receiver. Only the former knows the state of the world, whereas the latter does not. There are a number of states, each of which occurs with a certain possibility. After the sender finds out what the state of the world is, he sends a signal to the receiver from a set of possible signals. From this signal the receiver forms a prior belief about the real state of the world and takes this belief into account when offering a contract. A signaling game is a specific variant of a Bayesian game. Myerson (1979) introduced the Revelation Principle which states that any Bayesian Nash equilibrium of any Bayesian game can be represented by an incentive-compatible direct mechanism. This is a Bayesian game in which each player acts by sending a signal about his type and in which telling the truth results in a Nash equilibrium.

In equilibrium, both agents maximize their payoff, given the strategy of their opponent. Their payoffs depend on various parameters of the bonus-malus contract (e.g. base premium, the discounts in the different classes, deductible discount). This dependency is examined. The main focus of this thesis is to find optimal profit maximizing parameter values for the insurer, such that the insurer can separate the good from the bad drivers in equilibrium.

One of the most well known signaling games is The Market For Lemons by Akerlof (1970). In this game, the sender is selling a car to the receiver. The buyer is unaware of the car’s condition while the seller is. The seller can therefore send a signal about the condition of the car to the buyer by setting a high or a low price. Setting a high price sends the signal that the car is in a good state and a low price sends the signal that the car is in a less good state (a ”lemon”). Other well known papers in which signaling games are used are the Spence Education Model by Spence (1973) and the Wage-effort Hypothesis by Akerlof and Yellen (1990).

In this thesis, Contract Theory and the Principal-Agent Model are implemented to set up a signaling game between an individual and an insurer. Contract Theory provides the framework in which the insurer maximizes its expected profit in the presence of asymmetric information and incentives. This is explained in Laffont and Martimort (2002). In relation to the Principal-Agent Model, in this thesis, the insurer (I) is the principal and the policyholder (P ) is the agent. The agent can either be a good driver (θ1) or a bad driver (θ2) with respective probabilities ν and 1 − ν,

the insurer’s belief. In this model, the agent (the individual) first discovers his type. Secondly, the principal (the insurer) offers a contract. And finally, the agent accepts or refuses this contract.

This thesis handles a more enhanced version of this model, in which asymmetric information plays a role. The individual knows his type, but the insurer does not. The individual sends a signal about his type to the insurer. This signal, denoted by σ, can either have the value σ1 or σ2. In this

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case, σ is related to the individual’s decision to take or not to take a deductible. When he takes the deductible, he sends the signal σ1, i.e. the signal that he is a good driver, and in the latter case the

signal σ2, a bad driver, is send. Hereby, it is assumed that an individual opting for a deductible is a

good driver with a probability of at least 1/2, and vice versa. The underlying rationale behind this is as follows: a good driver does not want to pay a high premium for an insurance which he will not need regularly, whereas a bad driver does not want to pay a deductible every time he claims. In the case of a bonus-malus contract, good drivers are rewarded and bad drivers are penalized. This should automatically distribute the different types of drivers among the different classes.

This thesis starts with explaining the assumptions and basics of the model. Subsequently, the car insurance process is modeled as a signaling game. In order to illustrate the basics of the model, we assume the individual’s strategy is to determine its optimal critical claim size using a method based on Norman and Shearn (1980) and Dellaert et al. (1990). The policyholder calculates a value (based on his situation) beyond which he will always want to claim. This value is calculated using the present values of both claiming and not claiming. In the signaling game, the strategy of the individual is to decide whether or not he wants a deductible and whether or not he wants to be insured.

We start our analysis with a basic BM contract with two levels in which there is only one type of policyholder who is always insured. Section 2 explains the assumptions and basics of the model, on the basis of this basic contract. Subsequently, the possibility for the individual to reject the contract is implemented and the signaling game is introduced. Section 3 discusses the conditions for a separating equilibrium in which every type of individual takes the insurance and only the good drivers take a deductible. Given these conditions, optimal parameter values are found such that the insurer maximizes its expected profit. Section 4 considers a similar analysis for several pooling equilibria and compares these outcomes to the outcome in the separating equilibrium. We find that a separating equilibrium provides a higher expected profit for the insurer than a pooling equilibrium. This means that it is profitable for the insurer to separate the good from the bad drivers in equilibrium, as expected. In Section 5, the separating equilibrium from Section 3 is analyzed again when some assumptions are loosened. Section 6 explains a method that can be used to derive more general equilibrium outcomes in the signaling game. Finally, Section 7 discusses the most important conclusions from this thesis and suggests some recommendations for further research.

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2 The Model

2.1 Model Assumptions

In this section, the model and the assumptions behind the model are explained. Assume that there is one single policyholder who will always be insured and therefore does not make decisions on being insured or not. First, the insurer defines a bonus-malus contract with two classes. This entails setting the parameters πi (i = 1, 2), the premiums in class 1 and 2. Secondly, the policyholder

makes a damage with probability p and has no damage with probability q = 1 − p. Both the insurer and the insured are aware of this probability. When there is no damage, nothing happens. Whereas, when a damage does occur, the next step in the model is the policyholder’s choice to claim or not to claim his damage. This choice is rational on the basis of expected future payments regarding both claiming and not claiming. When he claims, he will always be in class 1 in the next period and when he does not claim, he will move to class 2. Note that this is independent of the class he is in right now. This is caused by the assumption that there are only two bonus-malus classes and makes calculations less complex.

The insurer and the insured are utility maximizers. Assume for simplicity that the agents are risk-neutral, i.e. monetary values as payoffs. To avoid any confusion, the payoffs are referred to by V or v (value). The policyholder’s payoff is dependent of his bonus-malus class, making or not making a damage and the claim decision. This leads to three different possible payoffs when there are two bonus-malus classes: the policyholder’s payoff when he has no damage, the payoff when he has a damage of size X and he decides to claim, and the payoff when he has a damage of size X and he decides not to claim. Here, X is a random variable representing the damage size, with density function fX.

The claim decision depends on the policyholder’s expected NPV. He chooses the option resulting in the highest NPV. Equation (1) represents the expected NPV of the policyholder in class i at time t. V_iE(t) = q · [πi+ V₂E(t + 1) 1 + r ] + p · min [ πi+ V₁E(t + 1) 1 + r ; πi+ V₂E(t + 1) 1 + r + X ], (1)

where V_iE(t) = E(Vi(t)), Vi(t) is the NPV of the total expected future payments as from t, when

the insured is in class i and r is the deterministic risk-free interest rate.

Say, x∗ is the critical value of X such that the policyholder is indifferent between claiming and not claiming. Then V1E(t+1)

1+r = VE

2 (t+1)

1+r + x

∗ _{should hold. This results in}

x∗ = V E 2 (t + 1) 1 + r − V₁E(t + 1) 1 + r . (2)

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Both the insurer and the insured are aware of the value of this optimal critical claim size x∗.

Depending on the time horizon, VE

i (t) can be calculated. It is assumed that the event of a

damage (with probability p) and the damage size (probabilities P(X ≥ x∗) and P(X ≤ x∗)) are independent. Therefore the probability that a damage occurs and the damage size is bigger than x∗ equals pP(X ≥ x∗). The NPV of the expected costs when the policyholder in class 1 is therefore: V₁E(t) = π1+q V₂E(t + 1) 1 + r +pE VE 1 (t + 1) 1 + r |X > x ∗ t P(X ≥ x∗)+pE V E 2 (t + 1) 1 + r + X|X ≤ x ∗ P(X ≤ x∗). (3)

When the horizon is only one time period, equation (3) becomes:

V₁E(t) = π1+ q π2 1 + r+ p π1 1 + rP(X ≥ x ∗ ) + p π1 1 + r + E (X|X ≤ x ∗ ) P(X ≤ x∗). (4)

and equation (2) becomes:

x∗ = π1 1 + r−

π2

1 + r. (5)

Assume π1 6= π2, i.e. 0 < a < 1. Note that x∗ does not depend on p.

All the assumptions are summarized in Table 1. ·There is only one policyholder.

·This policyholder does not make a decision on being insured or not: he always is

·The insured makes a damage with probability p

·Both the insurer and the insured know this probability.

·There are only 2 bonus-malus classes.

·The policyholder and the insurer are risk-neutral.

·The decision horizon of the policyholder is one period.

·The insurer has an effectively infinite horizon.

Table 1: summary of assumptions

Example 1

Equation (6) represents the expected value of the NPV in class 1, given at time t. Assume, for simplicity, that the policyholder’s decision horizon is one period. He does not take into account any of the payments after that period. Thus, the NPV consists of the premium he is paying right now, and the premium he expects to pay in the next period. In other words, only V_iE(0) and V_iE(1) are relevant. Equation (3) then results in:

V₁E(t) = π1+ q π2 1 + r + p π1 1 + r Z ∞ x∗t fX(x)dx + p π2 1 + r Z x∗_t 0 fX(x)dx + p Z x∗_t 0 XfX(x)dx, (6) whereRx∗t

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x∗_t, and thus, the policyholder is paying the damage himself. The integralRx∗_t

0 XfX(x)dx represents

the expected value of the damage given that it is below x∗_t. And R∞

x∗

t fX(x)dx is the probability

that the damage size is bigger than the critical claim size x∗_t, which means that the policyholder will claim the damage.

Assume that the damage size is exponentially distributed with λ = 1, such that fX = e−X. As

the cumulative distribution of fX is FX = 1 − e−X, the integral

R∞ x∗ t fX(x)dx results in e 1 1+r(π2−π1), the integral Rx∗t 0 fX(x)dx in 1 − e 1

1+r(π2−π1) _{and the integral} Rx ∗ t 0 XfX(x)dx = Rx∗_t 0 Xe −X_{dx =}

ke−X− 1 + e1+r1 (π2−π1)_{. When inserting these integrals, equation (5) simplifies to:}

V₁E(t) = π1+ q π2 1 + r+ p π1 1 + re 1 1+r(π2−π1)_{+ p} π2 1 + r 1 − e1+r1 (π2−π1) + pke−X − 1 + e1+r1 (π2−π1) , (7) which, when substituting q = 1 − p, results in:

V₁E(t) = π1+ π2 1 + r+ p π1 1 + re 1 1+r(π2−π1)_{+ p(ke}−X − 1). ₍₈₎

A similar argument leads to V₂E(t) = π2+ π2 1 + r + p π1 1 + re 1 1+r(π2−π1)_{+ p ke}−X − 1 . ₍₉₎

Note that only the first term is different.

2.2 Step 1: To Be or Not To Be Insured

In the previous subsection, the individual always took an insurance and therefore did not have the choice not to be insured. Consequently, the insurer does not need to take into account a possible termination of the contract and thus is able to set an infinitely high premium. Therefore, the first step from the simple model to the extended model is to include the individual’s strategic decision to be or not to be insured.

The policyholder makes this decision, given the premiums asked by the insurer and compares his expected payoffs of both being and not being insured. The payoff when not taking the insurance is as follows:

V_nE = p Z ∞

0

XfX(x)dx. (10)

Depending on the bonus-malus class the policyholder would enter when he takes the insurance, this payoff is compared to either V₁E or V₂E.

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nature 1 2 (V1,D(pg), v1,D(pg)) (Vno(pg), 0) (V2,D(pg), v2,D(pg)) (Vno(pg), 0) 2 (Vno(pg), 0) (V2(pg), v2(pg)) (Vno(pg), 0) (V1(pg), v1(pg)) 1 2 (V1,D(pb), v1,D(pb)) (Vno_(p b), 0) (V2,D(pb), v2,D(pb)) (Vno(pb), 0) 2 (Vno(pb), 0) (V2(pb), v2(pb)) (Vno_(p b), 0) (V1(pb), v1(pb)) p = pg p = pb D No D D No D BM1,D BM2,D BM2 BM1 BM1,D BM_2,D _BM 2 BM1 1 1 1 1 1 1 1 1 i n q 1 − q

Figure 1: The Game

The first agent is the individual who decides whether or not he would like a deductible (D or No D). The second agent is the insurer, who decides in which bonus-malus class (BM1 or BM2) he would like to place

the individual. In the last step, the individual decides whether or not he/she takes the insurance. This results in payoffs V for the individual and v for the insurer.

The premium in class 2 equals the premium in class 1 times a discount rate, i.e. π2 = a ∗ π1,

where a ∈ (0, 1). The insurer’s profit is maximized over the variable a, under the following condi-tion: V_nE(t−1) ≤ V2(t−1). This means that the individual’s expected costs of not having insurance

must be lower than or equal to the expected costs of having insurance. It is also assumed that the individual will always want to be insured when the expected costs of being insured are equal to the expected costs of not being insured. This restriction ensures that the insurer will not charge the maximal value of a, as otherwise, the individual does not buy the insurance.

Assumption: in case of equal expected payoffs, the individual will choose to be insured rather than not.

2.3 Step 2: Defining the Game

The final step is to define the signaling game. In this game, the possibility to choose a deductible is incorporated. When the individual decides to take a deductible, he receives a deductible discount. Figure 1 shows the game between the insurer and the individual. This is in essence an adapted version of a signaling game. First, nature randomly decides whether the individual is a good (with probability q) or a bad (with probability 1 − q) driver. This results in corresponding damage probabilities pg for a good driver and pb for a bad driver. The individual knows his type and the

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insurer does not. Secondly, the individual (player 1) decides whether or not he wants to bare the risk of having a deductible. This decision is an implicit signal to the insurer. Keeping this in mind, the individual can decide to act as if he is of a different type than he is as decided by nature. When the individual opts for a deductible, the insurer will lower his premium by B ∈ [0, π2]. This results

in the premiums π₁D = π1− B and π2D = a ∗ π1− B.

After the deductible decision of the individual, the insurer (player 2) decides whether it wants to place the individual in bonus-malus class 1 or 2. He makes this decision taking into account the signal he receives from the individual. The bonus-malus class is communicated to the individual, before he needs to make the decision whether or not to take the insurance. BM1,D and BM2,D

are the bonus-malus contracts which can be offered to the individual when he would like to have a deductible and BM1 and BM2 when he does not opt for a deductible.

After this step, an extra step is added, which makes this signaling game different from the common one, where the game ends after the decision of the second agent. In this extra step, the individual (player 1) decides whether or not he takes the insurance (denoted by no if he does not take the insurance). This last step results in the final payoffs for the insurer and the individual.

In Figure 1, V1,D, V1, V2,D and V2 stand for the individuals’ expected payoff when he is placed

in class 1 or 2 after choosing a deductible (D) or no deductible. v1,D, v1, v2,D and v2 represent

the insurer’s payoffs. When the individual does not take the insurance, the insurer’s payoff always equals 0. In Vi(p), i represents the bonus-malus class (i.e. 1 or 2) and p can either be pg or pb.

To calculate the payoffs in this game, the payoff formulas derived in Subsection 2.1 are used. Assume that the game is only played for one time step ahead. First, the individual’s payoffs are derived. If the individual decides not to take the insurance, the payoff will only depend on his damage probability. Since p and X are independent, its expected expenses are then defined by Vno(pg) = pgE[X] when he is a good driver or Vno(pb) = pbE[X] when he is a bad driver.

When the individual does take the insurance, the expected expenses are as in equation (11) or (12), depending on his deductible choice.

Without deductible: V_iE(t) = πi+ (1 − p) π2 1 + r + p " π1 1 + rP(X > x ∗ ) + π2 1 + rP(X 6 x ∗ ) + E[X|X 6 x∗] # , (11) where x∗ = π1 1+r − π2

1+r and p ∈ {pb, pg} as derived in equation (5). Recall that x

∗ _{is the critical}

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And with deductible: V_i,DE (t) = πi−B +(1−p) π2− B 1 + r +p " π1+ D − B 1 + r P(X > x ∗ D)+ π2− B 1 + r P(X 6 x ∗ D)+E[X|X 6 x ∗ D] # , (12) where x∗_D = D + x∗ = D + π1 1+r − π2

1+r. The rationale behind this is as follows: when the damage

size is lower than the deductible D, it is always more profitable for the individual not to claim his damage. It can easily be shown that V_1,DE (t + 1) − V_2,DE (t + 1) = V₁E(t + 1) − V₂E(t + 1) = π1

1+r− π2

1+r,

such that x∗_D = D + x∗ holds.

3 Separating Equilibrium Using Subgame Perfection

In the game defined in the previous section, multiple equilibria can arise. There can be pooling equilibria, separating equilibria or mixed strategy equilibria. There are two possible separating equilibria: one in which the good drivers do not take a deductible and the bad drivers do and one in which this is the other way around. This thesis focusses on finding the conditions for the latter separating equilibrium, a pooling equilibrium in which everyone takes a deductible and a pooling equilibrium in which no one takes a deductible. The conditions for these equilibria are derived using subgame perfection. In subgame perfect equilibria, every subgame of the game is in equilibrium. This means that in every part of the game, the agents have no incentive to change their strategy, given the strategy of the opponent. The three equilibria are compared on the basis of the insurer’s expected profit.

In this section, conditions for the subgame perfect separating equilibrium are derived, in which the individual will always opt for a deductible when he is a good driver and will never opt for a deductible when he is a bad driver. This equilibrium is of particular interest, as it is expected that the insurer ideally wants to use the deductible to separate the good from the bad drivers. Later, the justness of this expectation is checked by comparing the outcome of the separating equilibrium to outcome of a pooling equilibrium.

More specifically, we look at an equilibrium in which individuals in all parts of the game tree in Figure 1 wish to be insured. In order for this equilibrium to occur, the payoff of taking the insurance must always be at least the payoff of not taking the insurance. Therefore, the following two inequalities must hold: V1(pb) ≤ Vno(pb) and V1,D(pg) ≤ Vno(pg).

The inequality V1(pb) ≤ Vno(pb) implies:

π1+ (1 − p)_1+raπ1 + p " π1 1+rP(X > x ∗_{) +} aπ1 1+rP(X 6 x ∗_{) + E[X|X 6 x}∗_] # ≤ pE(X), which leads to the following condition on a:

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a ≤ 1 + 1 1 − pb· P(X > x∗) 1 + r π1 pb· E(X|X > x∗) − 2 − r . (13)

This is also an implicit condition on π1. Rewriting the above condition gives:

π1 ≤

(1 + r)pbE(X|X > x∗)

(a − 1) (P(X > x∗_{)) + 2 + r}. (14)

The inequality V1,D(pg) ≤ Vno(pg) implies:

π1− B + (1 − pg)a∗π_1+r1−B + p π1+D−B 1+r P(X > x ∗ D) +aπ1 −B 1+r P(X 6 x ∗ D) + E[X|X 6 x∗D] . This leads to a second condition on a:

a ≤ 1+(1 + r)pgE(X|X ≥ x ∗ D) + B (r(1 − pgP(X ≥ x∗D) + 2 + r) − DpgP(X ≥ x∗D) π1 1 − pgP(X ≥ x∗_D) − 2 + r 1 − pgP(X ≥ x∗_D). (15) Recall that x∗ and x∗_D do not depend on p!

When the aforementioned conditions hold and a ∈ (0, 1), the game in Figure 1 transforms into the game in Figure 2.

nature 1 2 (V1,D(pg), v1,D(pg)) (V2,D(pg), v2,D(pg)) 2 (V2(pg), v2(pg)) (V1(pg), v1(pg)) 1 2 (V1,D(pb), v1,D(pb)) (V2,D(pb), v2,D(pb)) 2 (V2(pb), v2(pb)) (V1(pb), v1(pb)) p = pg p = pb D No D D No D BM1,D BM2,D BM2 BM1 BM1,D BM2,D BM2 BM1 q 1 − q

Figure 2: The resulting game when the last step of the game in Figure 1 is solved using subgame perfection. The thick branches show the decisions made in the specific separating equilibrium in which good drivers opt for a deductible and bad drivers do not and both good and bad drivers take the insurance.

The insurer would like to place an individual with a deductible in class 2 and an individual without a deductible in class 1, as he assumes good drivers will opt for a deductible and bad drivers will not. In order for the latter to hold, for a bad driver, the costs of not taking a deductible must be lower than the costs of taking a deductible. And the other way around for a good driver. This is the case when V2,D(pg) ≤ V1(pg) and V2,D(pb) ≥ V1(pb) (see thick branches in Figure 2). These

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two inequalities can be rewritten to the following condition on B: Z1+ pbZ2 ≥ B ≥ Z1+ pgZ2, (16) where Z1 = 1+r_2+r(π2− π1) =1+r_2+rπ1(a − 1) and Z2 = P(x∗≤ X ≤ x∗D) π1(a−1) 2+r + 1+r 2+rE(X|x ∗ _{≤ X ≤ x}∗ D) +2+rD P(X ≥ x ∗ D).

In order to attain realistic values for B, B ≥ 0 must also hold.

To summarize, equation (13), (15) and (16) are the conditions that must hold in order for a separating equilibrium to occur in which the individual always wishes to be insured, and in which good drivers opt for a deductible and bad drivers do not. The derivation of these conditions can be found in the Appendix.

Example 2

In this example, assume the damage size to be log normally distributed with log(X) ∼ N(6.5,0.5). Thus the mean of the damages is assumed to be e6.5 ≈ 665 euros and about 68% of the damages lies between e6.5−0.5 _{≈ 403 and e}6.5+0.5 _{≈ 1097 euros. Assume that the rate of inflation r = 2%.}

Recall that x∗= π1

1+r− π2

1+r. Which becomes x ∗ ₌ 1

1.02(1 − a)π1. Take π1 = 70 as the base premium

per year, a = 0.33, the deductible D = 300, the discount when a policyholder has a deductible B = 5, pb = 0.35 and pg = 0.15. Then, x∗= _1.021 0.67 · 70 ≈ 47 and x∗_D = _1.021 0.67 · 70 + 300 ≈ 347.

Now we can calculate the probabilities and expectations. Table 2 shows all the values needed to find out whether or not the conditions for the separating equilibrium hold.

Table 2: The probabilities and expectations in the example

Now, condition (13) becomes a ≤ 0.3343 for p = 0.15 and a ≤ 3.1122 for p = 0.35 and (16) becomes −2.4475 ≤ B ≤ 25.8654 As a = 0.33 and B = 5, in this example, the desired separating equilibrium exist for the chosen parameter values. When π1 = 100 for example, the condition

a ≤ −0.1775 should hold. As a ∈ (0, 1), the separating equilibrium does not exist when we change π1 from 70 to 100 in this example.

In this particular equilibrium the expected profit of the insurer equals qv2,D(pg) + (1 − q)v1(pb).

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u(a, π1, B) = (1 − q + aq)π1 − qB − qpg(E(X|X ≥ x∗D) − D) − (1 − q)pbE(X|X ≥ x∗). (17)

Assuming the probability q that the individual is a good driver to be 0.75, gives an expected profit of approximately −82. This is a negative profit and thus, the insurer will not offer this contract when the parameter values are as in this example. The parameter values were chosen arbitrarily. Therefore, the profits will most likely be positive when other parameter values are chosen. The maximal expected profit for which the desired equilibrium occurs, can be calculated by performing a Lagrange optimization. The Lagrange optimization is done in Subsection 3.1.

3.1 Lagrange Optimization of the Separating Equilibrium

As x∗ and x∗_D depend on a non-linearly (a is in the integration boundaries), assume the threshold value to simply be a budget, ¯x, to make calculations less complex. If the damage size is bigger than this budget, the policyholder will always claim as he simply can not pay the damage himself. We maximize the insurer’s profit over a and B, keeping all the other parameters fixed. This includes π1, as Lagrange optimization including π1 becomes complex.

Assume, also for simplicity, that the distribution of X is as follows: a low damage (xl _{< ¯}_x)

occurs with probability σ and a high damage (xh > ¯x) occurs with probability 1 − σ. Therefore, P(x ≤ ¯x) = σ, E(X|X ≤ ¯x) = xl and E(X|X ≥ ¯x) = xh. The expected profit for the insurer becomes:

v(a, B) = π1(1 − q + qa) − qB − qpg

xh− D− (1 − q)p_bxh. (18)

The maximization of the insurer’s expected profit is done under the following conditions: a ≤ 1 ⇒ 1 − a ≥ 0

a ≤ f (¯x) ⇒ f (¯x) − a ≥ 0, which represents condition (13).

a ≤ g(B) ⇒ g(B) − a ≥ 0 Z(pg) ≤ B ≤ Z(pb) ⇒ Z(pb) − B ≥ 0 and B − Z(pg) ≥ 0, where f (¯x) = 1 + _1−p1 b(1−σ) 1+r π1 pb· x h_{− 2 − r}_, g(B) = 1 +(1+r)pgxh+B(r(1−pg(1−σ))+2+r)−Dpg(1−σ) π1(1−pg(1−σ)) − 2+r 1−pg(1−σ) and Z(p) = 1+r_2+r(a − 1)π1+ p ·_2+rD (1 − σ), with p ∈ {pg, pb}.

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Lagrange optimization of the expected profit under the previously mentioned conditions, leads to one feasible outcome. Namely, the outcome a = f (pb) and B = Z(pg, a). As the expression

for B depends on a, we need to substitute a in B before we get the final expressions. The final expressions for a and B are as follows:

a = f (pb) = 1 + (1 + r)pbxh π1(1 − pb(1 − σ)) − 2 + r 1 − pb(1 − σ) (19) and B = (1 + r) 2_p bxh (2 + r)(1 − pb(1 − σ)) + pg(1 − σ) D 2 + r− π1(1 + r) 1 − pb(1 − σ) . (20)

Substituting a and B in equation (18) gives:

v(a, B) = π1+ q(1 + r)(pbxh+ π1) − qπ1(2 + r) 1 − pb(1 − σ) − q(1 + r) 2_p bxh (2 + r)(1 − pb(1 − σ)) − pg(1 − σ)q D 2 + r − qpg(x h_{− D)} − (1 − q)p_bxh (21) The extensive Lagrange optimization can be found in the Appendix.

3.2 Separating Equilibrium: a Numerical Example

In this section, a numerical example of the Lagrange solution from the previous section is provided. This should provide a better understanding of the Lagrange outcome. Let r = 0.02, π1 = 180,

D = 300, pg = 0.15, pb = 0.35, xh = 1000, σ = 0.25 and q = 0.9. Then, a = 0.95, B = 12.19 and

v(a, B) = 31.48, the insurer’s expected profit. Now, it is interesting to see the effect of changing these parameter values. Table 3 shows the Lagrange solution for different values of the parameters. What can be deduced from Table 3 is that a higher q results in a higher profit. Since q represents the probability that the individual is a good driver and good drivers claim less often than bad drivers, this is a sensible result. Also, the higher σ, the higher the profit. σ is the probability that the damage size is lower than the critical claim size and therefore, it is also the probability that the policy holer will pay for the damage himself. The higher this probability, the less the insurer has to pay and therefore, the higher its profit.

The Lagrange outcome is very sensitive to changes in xh and pb. A little change in these

parameters, leads to infeasible outcomes. xh triggers all the costs the insurer has to make. The higher xh, the higher the insurer would like to set a and B, which finally leads to a > 1. The

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a B Feasible? Profit Profit > or <? q = 0.85 0.95 12.19 Yes 20.28 _≪ q = 0.95 0.95 12.19 Yes 42.67 ≫ σ = 0.2 0.95 13.19 Yes 30.38 < σ = 0.3 0.95 11.18 Yes 32.57 > xh = 900 0.68 −12.25 No 26.91 xh = 1100 1.22 36.63 No 36.04 pb = 0.3 0.59 −20.82 No 7.35 ≪ pb = 0.4 1.35 48.74 No 58.72 ≫ pg= 0.1 0.95 6.62 Yes 67.99 ≫ pg= 0.2 0.95 17.76 Yes −5.04 ≪ D = 250 0.95 9.40 Yes 27.23181 D = 350 0.95 14.97 Yes 35.71944 r = 0.015 0.9438795 11.66 Yes 30.91 < r = 0.025 0.9566855 12.72 Yes 32.03 > π1 = 175 1.027119 19.10 No 32.58 > π1 = 185 0.8775996 5.27 Yes 30.37 <

Table 3: The effect of changing one

of the fixed parameters (Ceteris Paribus) in the previously mentioned example. The first column represents the new value for one of the parameters, the second and third column show the resulting optimal values of a and B. The next column specifies whether a and B are feasible (i.e. B ≥ 0 and 0 ≤ a ≤ 1), the fifth column shows the resulting expected profit and in the last column it is specified whether this profit is higher or lower than the expected profit in the example. < or > represents a minor relative change, or a medium relative change and ≪ or ≫ a big relative change as opposed to the first example.

lower xh, the lower the insurer must set a and B, in order to keep the individual willing to take the insurance. Therefore B becomes negative. A similar reasoning holds for pb. Bad drivers are more

likely to claim than good drivers, since they do not have a deductible, and therefore, a higher pb

will trigger the insurer’s expenses to go up rather quickly.

In Table 3, we can also see that a decrease in pg leads to a decrease in B. A lower pg implies

that the policyholder will use his deductible less often and therefore, his expected costs of having a deductible are lower. Hence, in order to ensure that the good driver takes a deductible, a lower B is required than for a higher pg. Also, the lower pg, the higher the insurer’s profit. When pg

increases, the insurer’s profit quickly drops below zero.

When the deductible D decreases, the discount B and the profit decrease as well. The lower the deductible, the less costs are taken away from the insurer. Therefore, the policyholder receives less discount and the profit decreases. For an increase in the deductible this is the other way around. When the inflation rate increases, a, B and the profit increase as well. When inflation goes up, the insurer can ask a higher price for its product, which causes a and B to go up. This is also reflected in the profit, although it is only a small effect.

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A lower base premium π1, leads to a higher a and B. The first is explained by the fact that

a lower base premium results in a lower profit for the insurer. This is offset by setting a higher a. The discount B for having a deductible also becomes higher, as B depends positively on a. The increased profit is caused by the relative dependencies between the variables. When the base premium is lowered, the Lagrange outcome rapidly becomes infeasible, as a > 1. A vice versa reasoning holds for increasing π1. The base premium π1 can be increased up until 188, before

B < 0 which is infeasible.

4 Pooling Equilibrium Using Subgame Perfection

In order for a pooling equilibrium to occur in which at least the bad drivers want to be insured, the following inequalities must hold: V1,D(p) ≥ V1(p) and V2,D(p) ≥ V2(p) for p ∈ {pb, pg} and

V1,D(pb) ≥ Vno(pb). The assumptions are the same as in Subsection 3.1.

The insurer’s profit is maximized under the following restrictions: a ≤ 1 ⇒ 1 − a ≥ 0, a ≤ k(B) ⇒ k(B) − a ≥ 0 and B ≥ l(pb, a) ⇒ B − l(pb, a) ≥ 0, where a ≤ 1 + (1 + r)pbx h_{+ Br (1 − p} b(1 − σ) + 2 + r) − Dpb(1 − σ) π1(1 − pb(1 − σ)) − 2 + r 1 − pb(1 − σ) = k(B) (22) and B ≥ p D 2 + r(1 − σ) = l(p, a). (23)

There are two scenarios: one in which only the minimal condition V1,D(pb) ≥ Vno(pb) holds and

one in which also V1,D(pg) ≥ Vno(pg) holds, such that the good driver also takes the insurance. In

the first case, the expected profit equals:

vb(a, B) = (1 − q)

aπ1− B − pb(xh− D)

. (24)

Maximizing this profit under the above conditions, leads to two possible feasible outcomes. The first option is a = k(pb, B) and B = l(pb), which, substituting B in a gives the following final

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B = pb D 2 + r(1 − σ) (25) and a = 1 + (1 + r)pbxh+ pb(1 − σ)D r(1−pb(1−σ)) 2+r + r − 1 π1(1 − pb(1 − σ)) − 2 + r 1 − pb(1 − σ) . (26)

The second feasible option is a = 1 and B = l(pb). This solution is feasible as long as a ≤ k(pb, B)

holds. The resulting expected profit is trivially higher than for the first solution. Moreover, the latter solution would imply that there is no BM system, as a = 1. This is the optimal solution when k(pb, B) ≥ 1. When this condition is not satisfied, the first solution will be the outcome of

the Lagrange optimization.

When the condition V1,D(pg) ≥ Vno(pg) also holds, the good driver also takes the insurance.

This gives a different expected profit and therefore, a different Lagrange solution. In this case, the expected profit becomes:

vb,g(a, B) = aπ1− B − (qpg+ (1 − q)pb)(xh− D). (27)

Lagrange optimization of this profit, including the additional condition on a, leads to three possible feasible solutions. In all of these solutions B = l(pb) = pb(1 − σ)_2+rD .

Solution 1: a = k(pg, B) = 1 +

(1+r)pgxh+(1−σ)D(_2+rpbr(1−pg(1−σ)+2+r)−pg)

π1(1−pg(1−σ)) −

2+r 1−pg(1−σ)

This is the optimal solution when a ≤ k(pb) and a ≤ 1 also hold. I.e. k(pg) ≤ 1 and k(pg) ≤ k(pb).

Solution 2: a = k(pb, B) = 1 + (1+r)pbxh+pb(1−σ)D _{r(1−pb(1−σ))} 2+r +r−1 π1(1−pb(1−σ)) − 2+r 1−pb(1−σ)

That is feasible and gives the optimal solution when a ≤ k(pg, B) and a ≤ 1 also hold. I.e.

k(pb) ≤ k(pg) and k(pb) ≤ 1.

Solution 3: a = 1, which is feasible and gives the optimal solution when a ≤ k(pg, B) and

a ≤ k(pb, B) hold. I.e. k(p) ≥ 1 for both pb and pg.

The extensive Lagrange optimization of this pooling equilibrium can be found in Appendix ??.

4.1 Pooling Equilibrium: a Numerical Example

In this subsection, a numerical example for the pooling equilibrium is provided. This is done using the same values for the fixed parameters as in the numerical example for the separating equilibrium.

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As a reminder, we set r = 0.02, π1 = 180, D = 300, pg = 0.15, pb = 0.35, xh= 1000, σ = 0.25 and

q = 0.9. In the separating equilibrium this resulted in a = 0.95, B = 12.19 and an expected profit of v(a, B) = 31.48.

In the pooling equilibrium, Option 2 (i.e. a = k(pb, B) and B = l(pb)) from the Lagrange

optimization turns out to be the only feasible outcome in this example. When the good drivers do not participate, this results in a = 0.37, B = 38.99 and an expected profit of vb(a, B) = −21.68. This expected profit is higher than in the case that the good drivers are also insured. In order to ensure that the good drivers participate, a must be very low, which causes the insurer’s expected profit to drop. In the pooling equilibrium, it is therefore more profitable for the insurer to focus on the bad drivers. Moreover, for this specific example, the separating equilibrium results in a higher expected profit than the pooling equilibrium.

In this example, the expected profit in the pooling equilibrium is negative. This is caused by the specific parameter values that were chosen. When, for example, the parameter values of pb,

xh, σ and π1 are pb = 0.3, xh = 1500, σ = 0.8 and π1 = 400, the pooling equilibrium provides an

expected profit of vb(a, B) = 3.12, while still producing feasible outcomes for a and B. However, in order to compare the parameter dependencies of the outcomes in the separating equilibrium and the pooling equilibrium, the parameter values from the example in Subsection 3.2 are used.

Table 4 shows the effect of changing one of the parameters (ceteris paribus) on the equilibrium outcome in the pooling equilibrium.

The table shows that minor changes of the parameters in the pooling equilibrium never cause infeasibility in the Lagrange outcome. In the separating equilibrium this did occur when changing xh or pb (in both directions) and when decreasing π1. The outcome of the pooling equilibrium is

less sensitive for changes in the parameters than the outcome for the separating equilibrium. This is caused by the fact that the separating equilibrium requires more conditions than the pooling equilibrium.

Another difference between the pooling and separating outcome, is that the sign of the difference in expected profit when changing D (the deductible), is the other way around in the pooling equilibrium. In the separating equilibrium, a higher deductible caused the expected profit to increase. Here, the expected profit decreases when increasing the deductible. This means that, when D increases in the pooling equilibrium, the decrease in the insurer’s costs is higher than the increase of the discount B. Whereas, in the separating equilibrium this was the other way around. The parameter values which result in infeasible outcomes in the separating equilibrium, do result in feasible outcomes in the pooling equilibrium. Therefore, in some cases, the insurer must set the BM parameters such that a pooling equilibrium occurs. This holds for decreasing or increasing pb

and xhand decreasing π1. In all these cases, the expected profit in the pooling equilibrium is lower

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a B Feasible? Profit Profit > or <? q = 0.85 0.37 38.99 Yes −32.52 _≪ q = 0.95 0.37 38.99 Yes −10.84 ≫ σ = 0.2 0.32 41.58 Yes −22.93 σ = 0.3 0.43 36.39 Yes −20.48 xh = 900 0.10 38.99 Yes∗ −23.02 xh = 1100 0.64 38.99 Yes∗ −20.34 pb = 0.3 0.12 33.42 Yes∗ −22.24 < pb = 0.4 0.66 44.55 Yes∗ −20.62 > pg= 0.1 0.37 38.99 Yes −21.68 − pg= 0.2 0.37 38.99 Yes −21.68 − D = 250 0.47 32.49 Yes −21.05 >∗ D = 350 0.28 45.48 Yes −22.31 <∗ r = 0.015 0.36 39.08 Yes −21.88 < r = 0.025 0.38 38.89 Yes −21.48 > π1 = 175 0.43 38.99 Yes∗ −20.81 > π1 = 185 0.32 0.32 Yes −22.55 <

Table 4: Shows the effect of changing one of the fixed parameters (c.p.) in the example from Section 3.2. The setup of this table is the same as for Table 3. The ∗ indicates that this cell shows a difference with the separating equilibrium. For example, when π1 = 175 the column about feasibility says ’Yes∗’.

The ∗ indicates that for this same value of π1 in the separating equilibrium, the outcome was not feasible.

equilibrium occurs, as the separating equilibrium does not provide feasible outcomes.

One of the most important conclusions that can be drawn from comparison of Table 3 and Table 4, is that the separating equilibrium always provides a higher expected profit than pooling. Thus, it is profitable for the insurer to separate the good from the bad drivers.

It is also interesting to inspect what happens in case of a pooling equilibrium in which no one opts for a deductible. When all the fixed parameters are as previously mentioned, such equilibrium leads to the expected profit v(a) = −17.89, which is slightly lower than the expected profit in the pooling equilibrium where all players choose a deductible. This is the other way around when the policyholders are all placed in class 1 instead of class 2. Then, the expected profit without deductible becomes v(a) = −17.00 and with deductible v(a, B) = −10.40. This is caused by the fact that the optimal value of a is much higher in the pooling equilibrium without deductible than in the pooling equilibrium with deductible. See the Appendix for the Lagrange optimization of the pooling equilibrium without deductible.

When all the policyholders in the pooling equilibrium with deductible are placed in class 2, there are scenarios in which the pooling equilibrium provides a higher expected profit than the separating equilibrium, while the latter provides a feasible outcome as well. This occurs when q ≤ 0.55, ceteris paribus. When the policyholders are in class 1, this does not occur.

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5 Separating Equilibrium Without Budget Assumption

In this section, the insurer’s expected profit is maximized in the separating equilibrium without the assumption that the claim threshold value is a budget and without the assumption that the damage size can only be high or low with a given probability. As a is in the integration boundaries of all the damage size expectations and probabilities, this cannot be done analytically. Therefore, the maximal expected profit can only be found by numerical optimization. Recall that the following conditions must hold for the separating equilibrium to occur:

a ≤ 1 + (1+r)pbE(X|X>x∗) π1(1−pb·P(X>x∗)) − 2+r 1−pb·P(X>x∗), a ≤ 1 + (1+r)pgE(X|X≥x ∗ D)+B(r(1−pgP(X≥x ∗ D)+2+r)−DpgP(X≥x ∗ D) π1(1−pgP(X≥x∗D)) −_1−p 2+r gP(X≥x∗D) and B ≥ 1+r_2+rπ1(a − 1) + pg P(x∗ ≤ X ≤ x∗D) π1(a−1) 2+r + 1+r 2+rE(X|x ∗_{≤ X ≤ x}∗ D) +2+rD P(X ≥ x ∗ D) . All these constraints are written into an R program, which can be found in the Appendix. Take the same values for the parameters as in Section 3.2 and Section 4.1 (i.e. r = 0.02, π1 = 180,

D = 300, pg = 0.15, pb = 0.35, σ = 0.25 and q = 0.9) and assume that the damage size is log

normally distributed with log(X) ∼ N (log(2500),√0.2). This means that 68% of the damages lie between 2500 exp −2√0.2 ≈ 1022 and 2500 exp −2√0.2 ≈ 6115 assuming that this represents the average policyholder’s car. The resulting optimal values are a = 0.84 and B = 31. The corresponding expected profit for the insurer is v(a, B) = −258. With the budget assumption, this resulted in a = 0.95, B = 12.19 and the insurer’s expected profit v(a, B) = 31.48. The difference in expected profit is very high, but the profits are not comparable, because the damage size distribution and the average damage size are different. What we can compare is the effect of changing the fixed parameter values. Table 5 shows these effects when there is no budget assumption.

First of all, there is a difference in feasibility. Without the budget assumption, changes in pb

do not result in infeasible outcomes, whereas changes in pg do result in infeasible outcomes. This

is the other way around when the budget assumption is present. The effect of a change in π1 is

dependent on the presence of the budget assumption. If the budget assumption is present, a lower π1 leads to an infeasible outcome and if the budget assumption is not made, the outcome is feasible

and the insurer makes a higher profit.

Another difference between Table 3 and Table 5 is that the effect of changing pb or D is the

other way around. This is a sign that the budget assumption causes different outcomes. The profit is highly negative for all different parameter values in Table 5, which is caused by the high expected damage sizes and proportionally low π1. A π1 for which the optimal outcome is feasible

and non-negative can not be found by numerical analysis. A possible explanation for this is that we are now looking at a very specific equilibrium, in which we require all the individuals to take the

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a B Feasible? Profit Profit > or <? q = 0.85 0.84 31 Yes −282 _≪ q = 0.95 0.84 31 Yes −234 ≫ µ = log(2600) 0.94 39 Yes −266 µ = log(2400) 0.73 22 Yes −251 s =√0.3 0.84 30 Yes −257 > s =√0.1 0.84 32 Yes −259 < pb = 0.3 0.84 31 Yes∗ −246 ∗ pb = 0.4 0.84 31 Yes∗ −271 ∗ pg = 0.1 0.12 −44 No∗ −208 ≫ pg = 0.2 1.6 58 No∗ N aN ≪ D = 250 0.88 27 Yes −255 >∗ D = 350 0.79 34 Yes −262 <∗ r = 0.015 0.83 30 Yes −259 < r = 0.025 0.85 32 Yes −257 > π1= 175 0.9 36 Yes∗ −257.7 > π1= 185 0.76 25 Yes −260 <

Table 5: Shows the effect of changing one of the fixed parameters (c.p.) in the example. The setup of this table is the same as for Table 3. The ∗ indicates a difference with the separating equilibrium with the budget assumption. The NaN is explained by the fact that we chose for a lognormal distribution. When a > 1 (which also is infeasible), (1 − a) π1

1+r becomes negative and the log of a negative value does not exist.

insurance. It might be that it is more profitable when some individuals do not want to be insured. Moreover, the individual and the insurer are now only looking 1 time step ahead. Taking a longer horizon might produce different outcomes. Note that the latter two reasonings also hold when the budget assumption is made.

6 Deriving Equilibria

This section introduces a method to find other equilibria in the signaling game. Using this method all different possible equilibria of the game can be found. As a first step, we need to specify when the individual will and will not take the insurance. This choice, made in the last step of the tree, depends on the expected payoffs resulting from taking or not taking insurance. The main focus of this thesis is on comparing the separating and pooling equilibrium, as done in Section 3 and Section 4. Therefore, this section only shows the first step to find general equilibria in this game. The other steps, can be done in a similar fashion.

In the upper left part of the tree, shown in Figure 3, an individual who is placed in class 1 will take the insurance when pg· E[X] > V1,DE (t). Solving this inequality for B leads to condition (28).

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2 (U1,D(pg), u1,D(pg)) (Uno(pg), 0) (U2,D(pg), u2,D(pg)) (Uno_(p g), 0) D,pg BM1,D BM2,D i n 1 1

Figure 3: Upper left part of the game

B > 1 + r 2 + r π1+π2+pg π1+ D 1 + r P(X ≥ x ∗ D) + π2 1 + rP(X ≤ x ∗ D) − E(X|X ≥ x∗D) − π2 = b1(pg). (28) A similar reasoning, when the individual is placed in bonus-malus class 2, therefore leads to: B > 1 + r 2 + r 2π2+pg π1+ D 1 + r P(X ≥ x ∗ D) + π2 1 + rP(X ≤ x ∗ D)−E(X|X ≥ x ∗ D)−π2 = b2(pg). (29)

The derivation of both equation (28) and (29) can be found in the Appendix.

The only difference between equation (28) and equation (29) is that π1+ π2 in the first equation

is replaced by 2π2 in the latter equation. As π2 < π1, a1 > a2. This leads to decision Table 6

for the individual on the left part of the tree. The table reads as follows: when B ≤ b1(pg), the

individual will not take the insurance in both class 1 and class 2. When b1(pg) < B < b2(pg), the

individual will take the insurance when he is placed in class 1, but not when he is placed in class 2. When B ≥ b2(pg), the individual will take the insurance in both class 1 and 2.

B ≤ b1(pg) b1(pg) < B < b2(pg) B ≥ b2(pg)

(No,No) (Yes,No) (Yes,Yes)

Table 6: individual’s decision on the upper left part of the tree. The first entry of (·, ·) corresponds to the decision of the individual when he is in class 1 and the second entry to his decision when he is in class 2.

For the bottom left part of the tree, shown in Figure 6, the conditions and decision table look very similar. Only, pg is replaced pb.

In the bottom left part of the tree, as displayed in Figure 4, the individual takes the insurance when pbE[X] ≥ V1,DE (t). This leads to conditions (30) and (31) (see Appendix for derivation) and

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(U1,D(pb), u1,D(pb)) (Uno(pb), 0) (U2,D(pb), u2,D(pb)) (Uno_(p b), 0) D,pb BM1,D BM2,D 1 1 i n

Figure 4: The bottom left part of the game

B > 1 + r 2 + r π1+ π2+ pb π1+ D 1 + r P(X ≥ x ∗ D) + π2 1 + rP(X ≤ x ∗ D) − E(X|X ≥ x∗D) − π2 = b1(pb) (30) B > 1 + r 2 + r 2π2+ pb π1+ D 1 + r P(X ≥ x ∗ D) + π2 1 + rP(X ≤ x ∗ D) − E(X|X ≥ x ∗ D) − π2 = b2(pb) (31) B ≤ b1(pb) b1(pb) < B < b2(pb) B ≥ b2(pb)

(No,No) (Yes,No) (Yes,Yes)

Table 7: individual’s decision on the bottom left part of the tree

For the right part of the tree, the conditions are different, as B is not present in the payoffs there. First, for the upper right part of the tree:

2 (Uno_(p g), 0) (U2(pg), u2(pg)) (Uno(pg), 0) (U1(pg), u1(pg)) No D,pg BM2 BM1 1 1 i n

Figure 5: The upper right part of the game

In the case where the individual is placed in class i pgE[X] ≥ ViE(t) must hold. This leads to

similar conditions as in the left part of the tree. See the Appendix for a derivation.

a ≤ 1 1 − pgP(P ≥ x∗) 1 + r π1 pgE(X|X ≥ x∗) − 1 − r − pg = a1(pg) (32)

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a ≤ 1 2 + r − pgP(X ≥ x∗) 1 + r π1 pgE(X|X ≥ x∗) − pgP(X ≥ x∗) = a2(pg) (33)

Conditions (32) and (33) are combined in decision Table 8. a ≤ a1(pg) a1(pg) < a < a2(pg) a ≥ a2(pg)

(Yes,Yes) (No,Yes) (No,No)

Table 8: individual’s decision on the upper right part of the tree

In the bottom right part of the tree, the condition pbE[X] ≥ ViE(t) must hold.

2 (Uno(pb), 0) (U2(pb), u2(pb)) (Uno_(p b), 0) (U1(pb), u1(pb)) No D,pb BM2 BM1 1 1 i n

Figure 6: The bottom right part of the game

This leads to the following conditions on a:

a ≤ 1 1 − pbP(P ≥ x∗) 1 + r π1 pbE(X|X ≥ x∗) − 1 − r − pb = a1(pb) (34) a ≤ 1 2 + r − pbP(X ≥ x∗) 1 + r π1 pbE(X|X ≥ x∗) − pbP(X ≥ x∗) = a2(pb) (35) a ≤ a1(pb) a1(pb) < a < a2(pb) a ≥ a2(pb)

(Yes,Yes) (No,Yes) (No,No)

Table 9: individual’s decision on the bottom right part of the tree

Decision tables similar to the ones derived for the insurance or no insurance decision can be derived for the insurer’s decision to place the individual in BM class 1 or 2 and for the individual’s decision to take or not to take a deductible. When decision tables are derived for all steps in the game, general equilibria can be derived.

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7 Conclusions & Recommendations

Various papers have been written about optimal parameters in bonus-malus contracts. This thesis, however, specifies a bonus-malus contract as a signaling game, which has never been done before. Moreover, an extra step is added to the common signaling game. In this step, the individual has the choice to refuse the contract.

The main question of this thesis was: What are the conditions for a separating equilibrium in which good drivers opt for a deductible and bad drivers do not? In order for the aforementioned equilibrium to exist, conditions on bonus-malus discount a and the deductible discount B were derived. Under these conditions, a Lagrange optimization of the insurer’s expected premium was performed. The resulting optimal values of a and B were the largest allowable value of a respectively the smallest allowable value of B.

A similar Lagrange optimization was performed for three pooling equilibria. In two of these, the insured drivers opt for a deductible: one in which only the bad drivers want to be insured and one in which both the good and the bad drivers want to be insured. The latter provides a lower expected profit for the insurer than the former, as a stricter condition on the premiums is necessary in order to ensure that the good drivers take the insurance as well. In the third pooling equilibrium, only the bad drivers are insured and they do not opt for a deductible. The pooling equilibrium in which only the bad drivers want to be insured and opt for a deductible, provides the highest expected profit of all three pooling equilibria. Therefore, this pooling equilibrium was compared to the aforementioned separating equilibrium. The most important conclusion from that comparison is that the separating equilibrium results in a higher expected profit than the pooling equilibrium, for all different parameter values, as expected. On the other hand, there are some scenarios in which the separating equilibrium does not provide feasible optimal outcomes for a and B, and thus, the insurer must set a and B such that the pooling equilibrium occurs. This happens for certain values of xh, pb and π1.

The numerical analysis of the separating equilibrium led to a highly negative expected profit for the insurer, which is caused by a low π1. When increasing π1, the optimal values for a and

B quickly become infeasible. This is due to the risk neutrality assumption and the restricted horizon of 1 period. When individuals are risk neutral, they are less likely to buy insurance. In the real world, however, individuals are risk averse, so that they will buy insurance more quickly. Therefore, there will most likely be more feasible outcomes with a higher expected profit when risk averseness is taken into account by using utility functions. Also, the very specific equilibrium in which all individuals take the insurance, forces π1 to be low. Otherwise, good drivers will not take

the insurance. Most probably, loosening the assumption that all individuals buy insurance will therefore lead to a higher expected profit.

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7.1 Recommendations

There are some assumptions which were made for simplicity. It can be interesting to loosen some of them and find out how the equilibrium outcomes change. Also, some possible improvements to this research are addressed.

In this thesis, individuals are risk neutral. However, it is assumed that when the expected costs of having and not having insurance are equal, the individual will always take the insurance. Nonetheless, there is no real risk averseness incorporated in the model. It is interesting to find out how the optimal equilibrium outcomes change when risk averseness is taken into account. Risk averseness will most likely cause the individual to take the insurance more quickly, i.e. for higher values of π1.

The current game only has 2 bonus-malus classes. As most car insurance firms in the Netherlands have around 15 to 20 classes, it is interesting to adjust the contract to a bigger amount of classes. The main idea of the model will not change, but more types of drivers will be needed. Namely, adding an extra class will not add extra value to the insurer when he only distinguishes 2 different types of drivers.

Finally, taking a longer horizon than 1 year can produce different outcomes. When the horizon is longer, there are more factors that play a role. Namely, when there are multiple bonus-malus classes, it is more interesting for a good driver to be insured, even though he will pay a higher premium in the first few years of his insurance. A longer horizon might therefore cause good drivers to take the insurance more quickly, which in turn allows the insurer to set higher premiums and make more profit. Further research on longer horizons can be interesting to find out whether this is the case.

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References

Akerlof, G. (1970). The market for ”lemons”: Quality uncertainty and the market mechanism. The Quarterly Journal of Economics, 84, 488–500.

Akerlof, G. and Yellen, J. (1990). The fair wage-effort hypothesis and unemployment. The Quarterly Journal of Economics, 105, 255–283.

Dellaert, N.P., Frenk, J.B.G., Kouwenhoven, A. and Van der Laan, B.S. (1990). Optimal claim behaviour for third party liability insurance or to claim or not to claim: that is the question. Insurance: Mathematics and Economics, 9, 59–76.

Gomez, E., Hernández, A., Péreza, J.M. and Vázquez-Polo, F.J. (2002). Measuring sensitivity in a bonusmalus system. Insurance: Mathematics and Economics, 31, 105–113.

Kliger, D. and Levikson, B. (2002). Pricing no claims discount systems. Insurance: Mathematics and economics, 31, 191–204.

Laffont, J. and Martimort, D. (2002). The Theory of Incentives: The Principal-Agent Model. Princeton University Press.

Majeske, K.D. (2007). A non-homogeneous poisson process predictive model for automobile war-ranty claims. Reliability Engineering and System Safety, 92, 243–251.

Myerson, R. (1979). Incentive compatibility and the bargaining problem. Econometrica, 47, 61–74. Norman, J.M. and Shearn, D.C.S. (1980). Optimal claiming on vehicle insurance revisited. The

Journal of the Operational Research Society, 31, 181–186.

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A

Appendix

A1. Derivation of equation (13)

In the desired separating equilibrium, the good drivers take a deductible and the bad drivers do not. To ensure that the individual takes the insurance in every part of the game, the expected payoff of not taking the insurance must be lower than or equal to the expected payoff of taking the insurance. In the case of the bad driver this results in the following:

V1(pb) ≤ Vno(pb) ⇒ π1+ (1 − pb)a·π_1+r1 + pb " π1 1+rP(X > x ∗_{) +} a·π1 1+rP(X 6 x ∗_{) + E[X|X 6 x}∗_] # ≤ p_b_{· E(X)} aΠ1 1+r 1 − pb+ pbP(X ≤ x ∗_{) ≤ p} b E(X) − E(X|X ≤ x∗) − π1 1 +pbP(X≥x ∗₎ 1+r ⇒ a 1 − pbP(X ≥ x∗) ≤ 1+r_Π₁ pb E(X) − E(X|X ≤ x∗) − r − 1 − pbP(X ≥ x∗) ⇒ a ≤ 1+r Π1 1−pbP(X≥x∗) pb E(X) − E(X|X ≤ x∗) + 1 −_1−p_b2+r_P(X≥x∗ ⇒ a ≤ 1 + _1−p 1 bP(X≥x∗) 1+r π1 pbE(X|X ≥ x ∗_{) − 2 − r}

is the condition that should hold to require that taking the insurance without a deductible is more profitable for the bad driver than not tak-ing an insurance.

A2. Derivation of condition (15)

To ensure that the good driver takes the insurance, the expected payoff of taking the insurance with a deductible must be higher than or equal to the expected payoff of not taking insurance. This results in:

V1,D(pg) ≤ Vno(pg) ⇒ π1− B + (1 − pg)a∗π_1+r1−B + pg π1+D−B 1+r P(X > x ∗ D) +aπ1 −B 1+r P(X 6 x ∗ D) + E[X|X 6 x∗D]

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⇒ aπ1 1+r(1 − pg+ pgP(X ≤ x ∗ D)) ≤ pgE(X|X ≤ x∗D)+B−π1−pgπ1+D−B_1+r P(X ≥ x∗D)+B (1 − pg+ pgP(X ≤ x∗D)) ⇒ a π1 1+r(1 − pgP(X ≥ x ∗ D)) ≤ pgE(X|X ≥ x ∗ D) − π1 + B (2 − pgP(X ≥ x ∗ D)) − pg π1+D−B 1+r P(X ≥ x ∗ D) ⇒ a ≤ (1+r) pgE(X|X≥x∗D)−π1+B(2−pgP(X≥x ∗ D))−pgπ1+D−B1+r P(X≥x ∗ D) π1(1−pgP(X≥x∗D)) ⇒ a ≤ (1+r)pgE(X|X≥x ∗ D)+B((1+r)(2−pgP(X≥x ∗ D))+pgP(X≥x ∗ D))−DpgP(X≥x ∗ D) π1(1−pgP(X≥x∗D)) −_1−p 1+r gP(X≥x∗D) − pgP(X≥x∗D) 1−pgP(X≥x∗D) As −_1−p 1+r gP(X≥x∗D) − pgP(X≥x∗D) 1−pgP(X≥x∗D) = − 1+r+pgP(X≥x∗D) 1−pgP(X≥x∗D) = 1 − 2+r 1−pgP(X≥x∗D) ⇒ a ≤ 1 +(1+r)pgE(X|X≥x ∗ D)+B((1+r)(2−pgP(X≥x∗D))+pgP(X≥x∗D))−DpgP(X≥x∗D) π1(1−pgP(X≥x∗D)) − 2+r 1−pgP(X≥x∗D) a ≤ 1 + (1+r)pgE(X|X≥x ∗ D)+B(r(1−pgP(X≥x ∗ D)+2+r)−DpgP(X≥x ∗ D) π1(1−pgP(X≥x∗D)) − 2+r

1−pgP(X≥x∗D) is the condition that

should hold to require that the good drivers take the insurance with a deductible.

A3. Derivation of condition (16)

The following two conditions must hold in order for the desired separating equilibrium to ex-ist: V2,D(pg) ≤ V1(pg) and V2,D(pb) ≥ V1(pb). We first rewrite both to a condition of B.

V2,D(pg) ≤ V1(pg) ⇒ π2− B + (1 − pg)Π_1+r2−B + pg Π1+D−B 1+r P(X ≥ x ∗ D) +Π2 −B 1+r P(X ≤ x ∗ D) + E(X|X ≤ x∗D) ≤ π₁+ (1 − pg)_1+rΠ2 + p Π1 1+rP(X ≥ x ∗_{) +} Π2 1+rP(X ≤ x ∗_{) + E(X|X ≤ x}∗₎ ⇒

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−B 1 +1−pg 1+r + pg 1+rP(X ≥ x ∗ D) + pg 1+rP(P ≤ x ∗ D) ≤ π1− π2+ pg Π1 1+r(P(X ≥ x ∗ ) − P(X ≥ x∗D)) + Π2 1+r(P(X ≤ x ∗ ) − P(X ≤ x∗D)) + E(X|X ≤ x ∗ ) − E(X|X ≤ x∗D) − D 1+rP(X ≥ x ∗ D) ⇒ −2+r 1+rB ≤ π1− π2+ pg P(x∗≤ X ≤ x∗D)Π1 −Π2 1+r − E(X|x ∗_{≤ X ≤ x}∗ D) −1+rD P(X ≥ x ∗ D) ⇒ B ≥ 1+r_2+r(π2− π1) + pgP(x∗≤ X ≤ x∗D) Π2−Π1 2+r + pg 1+r 2+rE(X|x ∗_{≤ X ≤ x}∗ D) + pg D 2+rP(X ≥ x ∗ D)

Rewriting the second inequality gives: V2,D(pb) ≥ V1(pb) ⇒ π2− B + (1 − pb)Π_1+r2−B + pb Π1+D−B 1+r P(X ≥ x ∗ D) + Π2−B 1+r P(X ≤ x ∗ D) + E(X|X ≤ x ∗ D) ≥ π1+ (1 − pb)_1+rΠ2 + pb Π1 1+rP(X ≥ x ∗_{) +} Π2 1+rP(X ≤ x ∗ ) + E(X|X ≤ x∗) ⇒ −B 1 +1−pb 1+r + pb 1+rP(X ≥ x ∗ D) + pb 1+rP(P ≤ x ∗ D) ≥ π1− π2+ pb Π1 1+r(P(X ≥ x ∗_{) − P(X ≥ x}∗ D)) + Π2 1+r(P(X ≤ x ∗ ) − P(X ≤ x∗D)) + E(X|X ≤ x ∗ ) − E(X|X ≤ x∗D) − 1+rD P(X ≥ x ∗ D) ⇒ −2+r_1+rB ≥ π1− π2+ pb P(x∗≤ X ≤ x∗D)Π1 −Π2 1+r − E(X|x ∗ _{≤ X ≤ x}∗ D) −1+rD P(X ≥ x ∗ D) ⇒ B ≤ 1+r_2+r(π2− π1) + pbP(x∗≤ X ≤ x∗D) Π2−Π1 2+r + pb 1+r 2+rE(X|x ∗_{≤ X ≤ x}∗ D) + pb_2+rD P(X ≥ x∗D)

Combining these two leads to the following condition:

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where Z1 = 1+r_2+r(π2− π1) and Z2 = P(x∗≤ X ≤ x∗D) Π2−Π1 2+r + 1+r 2+rE(X|x ∗ _{≤ X ≤ x}∗ D) +2+rD P(X ≥ x ∗ D)

A4. Derivation of equation (46)

Vi,D(p) ≤ Vi(p) ⇒ πi− B + (1 − p)aπ_1+r1−B + p π1+D−B 1+r P(X ≥ x ∗ D) + aπ1−B 1+r P(X ≤ x ∗ D) + E(X|X ≤ x∗D) ≤ π_i+ (1 − p)aπ1 1+r+ p π1 1+rP(X ≥ x ∗_{) +} aπ1 1+rP(X ≤ x ∗ ) + E(X|X ≤ x∗) ⇒ −B − (1 − p)_1+rB + pD−B_1+r P(X ≥ x∗_D) − p_1+rB P(X ≤ x∗_D) ≤ p_1+rπ1 (P(X ≥ x∗) − P(X ≥ x∗_D)) + paπ1 1+r(P(X ≤ x ∗ ) − P(X ≤ x∗D)) + p (E(X|X ≤ x ∗ ) − E(X|X ≤ x∗D)) ⇒ −B2+r_1+r+ p_1+rD P(X ≥ x∗D) ≤ pP(x∗ ≤ X ≤ x∗D)(1 − a)1+rπ1 − pE(X|x ∗_{≤ X ≤ x}∗ D) ⇒ −B2+r 1+r ≤ pP(x ∗_{≤ X ≤ x}∗ D)(1 − a) π1 1+r − pE(X|x ∗ _{≤ X ≤ x}∗ D) − p1+rD P(X ≥ x ∗ D) ≤ pP(x∗ ≤ X ≤ x∗_D)(1 − a) π1 1+r + pE(X|x ∗_{≤ X ≤ x}∗ D) ⇒ B ≥ p1+r_2+rE(X|x∗≤ X ≤ x∗D) +2+rD P(X ≥ x ∗ D) − (1 − a) π1 2+rP(x ∗_{≤ X ≤ x}∗ D)

A5. Lagrange in the Separating Equilibrium

As x∗ and x∗_D depend on a in a non-linear fashion (a is in the integration boundaries), assume the threshold value to simply be a budget, ¯x. If the damage size is bigger than this budget, the policyholder will always claim as he simply can not pay the damage itself. The insurer’s profit is maximized over a and B, keeping all the other parameters fixed. This includes π1, as Lagrange

optimization including π1 makes calculations too complex.

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a ≤ 1 ⇒ 1 − a ≥ 0

a ≤ f (¯x) ⇒ f (¯x) − a ≥ 0, which represents condition (13) and therefore

f (¯x) = 1 + 1 1 − pb· P(X > ¯x) 1 + r π1 pbE(X|X > ¯x) − 2 − r , (36) a ≤ g(B) ⇒ g(B) − a ≥ 0, with g(B) = 1 + (1+r)pgE(X|X≥¯x)+B(r(1−pgP(X≥¯x)+2+r)−DpgP(X≥¯x) π1(1−pgP(X≥¯x)) − 2+r 1−pgP(X≥¯x). Z(pg) ≤ B ≤ Z(pb) ⇒ Z(pb) − B ≥ 0 and B − Z(pg) ≥ 0, where Z(p) = 1+r_2+r(a − 1)π1+ p ·_2+rD P(X ≥ ¯x) with p ∈ {pg, pb}.

The restriction a ≥ 0 is purposely left out, as the higher a, the higher the insurer’s profit. Assume, also for simplicity, that the distribution of X is as follows: a low damage, xl < ¯x, with probability σ and a high damage, xh > ¯x, with probability 1 − σ. Such that P(x ≤ ¯x) = σ, E(X|X ≤ ¯x) = xl and E(X|X ≥ ¯x) = xh.

Now, f (¯x), g(B) and Z(p) can be rewritten to:

f (¯x) = 1 + 1 1 − pb(1 − σ) 1 + r π1 pb· xh− 2 − r , (37) g(B) = 1 + (1 + r)pgx h_{+ B (r(1 − p} g(1 − σ)) + 2 + r) − Dpg(1 − σ) π1(1 − pg(1 − σ)) − 2 + r 1 − pg(1 − σ) (38) and Z(p) = 1 + r 2 + r(a − 1)π1+ p · D 2 + r(1 − σ). (39)

Without the above simplification, the insurer’s profit is:

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This becomes:

v(a, B) = π1(1 − q + qa) − qB − qpg

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The resulting Lagrangian:

L(a, B, µ) = v(a, B)+µ1(1−a)+µ2(f (¯x)−a)+µ3(g(B)−a)+µ4(Z(pb)−B)+µ5(B −Z(pg)), (42)

where µ = (µ1, ..., µ5).

The derivatives of this Lagrangian are: ∂L ∂B = −q + µ3 dg(B) dB − µ4+ µ5 = −q + µ3 r(1 − pg(1 − σ)) + 2 + r π₁2(1 − pg(1 − σ)) − µ₄+ µ5 = 0 (43) and ∂L ∂a = qπ1− µ1− µ2− µ3+ µ4 dZ(pb) da − µ5 dZ(pg) da = qπ1− µ1− µ2− µ3+ (µ4− µ5)π1 1 + r 2 + r = 0. (44)

µ1, µ2 and µ3 are all linked to conditions on a. This means that at least two of these should

equal zero. I assume that µ1 = 0, as a = 1 is an undesirable outcome. This leads to the following

three options: µ1 = µ2 = µ3 = 0, µ1 = µ2 = 0 and µ1 = µ3 = 0. A similar reasoning for µ4 and

µ5, which are both related to B, leads to the three options µ4 = µ5 = 0, µ4 = 0 and µ5 = 0. This

means that there are 3 · 3 = 9 possibilities. From these possibilities, a few can already be ruled out. From the derivative _δBδL, it can be deduced that µ3, µ4 and µ5 cannot all be zero at the same time,

as this would require q = 0. Possibilities in which µ3 = µ5 = 0 can also be ruled out, as this would

require µ4 = −q < 0. This leaves us with 5 possibilities which should be checked.

Option 1: µ1 = µ2= µ3 = µ4 = 0, such that B = Z(pg). ∂L ∂B = −q + µ5 = 0 ⇒ µ5 = q. ∂L ∂a = qπ1− qπ1 1+r 2+r = qπ1 2+r = 0. So, ∂L

∂a = 0 is infeasible since q, π1 6= 0.

Option 2: µ1 = µ2= µ4 = µ5 = 0, such that a = g(B). ∂L ∂B = −q + µ3 r(1−pg(1−σ))+2+r π2 1(1−pg(1−σ)) = 0 ⇒ µ3= q π2 1(1−pg(1−σ)) r(1−pg(1−σ))+2+r. ∂L ∂a = qπ1− q π21(1−pg(1−σ))

r(1−pg(1−σ))+2+r = 0, which does not lead to a solution for B and is thus infeasible.

Option 3: µ1 = µ2= µ4 = 0, such that a = g(B) andB = Z(pg). ∂L ∂B = −q + µ3 r(1−pg(1−σ))+2+r π2 1(1−pg(1−σ)) + µ5 = 0 ⇒ µ5= q − µ3 r(1−pg(1−σ))+2+r π2 1(1−pg(1−σ)) . ∂L ∂a = qπ1−µ3− q − µ3r(1−p_π2 g(1−σ))+2+r 1(1−pg(1−σ)) π11+r_2+r = 0 ⇒ _2+rqπ1+µ3(1+r)(r(1−pg(1−σ))+2+r)−(2+r)π_(2+r)(1−p_g_(1−σ)) 1(1−pg(1−σ)) =

Estimating an optimal bonus-malus contract with asymmetric information.