Problem Set 4 - Solution Hints

Transcription

1 ETH Zurich D-MTEC Chair of Risk & Insurance Economics (Prof. Mimra) Exercise Class Spring 206 Anastasia Sycheva Contact: Office Hour: on appointment Zürichbergstrasse 8 / ZUE, Room F2 Problem Set 4 - Solution Hints. Background Risk a.) Some reasons behind uninsurability of risks might be:.) High correlation. Example: catastrophic risks. 2.) High transaction costs that include but are not limited to administrative costs linked to the monitoring of individual insurance policies. These costs yield additional loading factor for insurance pricing, which in some cases exceeds the customer s risk premium 3.) Adverse Selection. Example: unemployment risks, health risks 4.) Moral Hazard. Example: failure at school or university, divorce, lack of demand for a new product 5.) Ambiguity. Expamle: global warming 6.) The absence of enough information about the risk may induce consumers and insurers to have different opinions about the intensity of the risk. The list above is not comprehensive. For a more extensive treatment of this subject I refer the reader to Gollier (2004) b.) The agent s wealth in four possible states of the world is given in Table Loss 0 L 0 W = W 0 pc W 3 = W 0 L + ( p)c D W 2 = W 0 pc D W 4 = W 0 L + ( p)c D Table : State contingent wealths of the individual. For the probabilities of the states f i since they are mutually exclusive and exhaustive it holds fi =. The exact expression depends on the nature of the stochastic relationship between the losses.

2 .) The insurable losses L and uninsurable losses D are statistically independent f = ( π)( τ), f 3 = π( τ), f 2 = ( π)τ, f 4 = πτ 2.) The insurable losses L and uninsurable losses D are perfectly positively correlated f = ( π) = ( τ), f 4 = π = τ f 3 = f 2 = 0 3.) The insurable losses L and uninsurable losses D are perfectly negatively correlated (π = τ) f = f 4 = 0 f 2 = τ, f 3 = π The buyer s maximization problem with respect to the optimal cover C an be written as follows: [ 4 ] max C f s u(w s ) = s= max C [f u(w 0 pc) + f 2 u(w 0 pc D) + f 3 u(w 0 L + ( p)c) + f 4 u(w 0 L + ( p)c D)] We differentiate the expression above with respect to C to obtain the First Order Conditions: p[f u (W ) + f 2 u (W 2 )] + ( p)[f 3 u (W 3 ) + f 4 u (W 4 )] = 0 f u (W ) + f 2 u (W 2 ) f 3 u (W 3 ) + f 4 u (W 4 ) = p p c.) Substitute the expressions for f i, i =,.., 4 from task a.) and p = π into the First Order Conditions to get: ( π)u (W ) πu (W 4 ) Since u(w ) is a concave utility function we have: = π π u (W ) u (W 4 ) = W = W 4 W 0 pc = W 0 L + ( p)c D C = L + D d.) Substitute the expressions for f i, i =,.., 4 from task a.) and p = π into the First Order Conditions to get: π( p) p( π) = ( τ)u (W ) + τu (W 2 ) ( τ)u (W 3 ) + τu (W 4 ) = Note that W W 3 = W 2 W 4 = C L. Since u < 0 we deduce that: If C < L u (W ) > u (W 3 ), u (W 2 ) > u (W 4 ) ( τ)u (W ) + τu (W 2 ) ( τ)u (W 3 ) + τu (W 4 ) > If C > L u (W ) < u (W 3 ), u (W 2 ) < u (W 4 ) ( τ)u (W ) + τu (W 2 ) ( τ)u (W 3 ) + τu (W 4 ) < If C = L u (W ) = u (W 3 ), u (W 2 ) = u (W 4 ) ( τ)u (W ) + τu (W 2 ) ( τ)u (W 3 ) + τu (W 4 ) = Thus we have proved the assertion. 2

3 e.) Substitute the expressions for f i, i =,.., 4 for negatively correlated losses from task a.) and p = π into the First Order Conditions to get: ( π)u (W 0 pc K) πu (W 0 L + ( p)c) = p p u (W 0 pc D) = u (W 0 L + ( p)c) Since u(w ) is a concave utility function the conditions above imply that: W 0 pc D = W 0 L + ( p)c C = L D. Thus in order to equalize incomes across states cover has to be bought which just makes up the difference between L and D. The uninsurable loss D plays a role of a natural hedge in the sense that with its existence less insurance is needed to achieve equal payoffs in both states of the world than if it did not exist. 2. Limitations of Expected Utility a.) Def Preference relation is a binary relation on the set of alternatives X, allowing the comparison of pairs of alternatives x, y X Def The preference relation is rational if it possesses the following properties: Completeness: for all x, y X we have x y or y x (or both). Problem: It is hard to evaluate alternatives that are far from the realm of common experience Transitivity: for all x, y, z X if x y and y z then x z Problem: Transitivity assumption is also hard to satisfy when evaluating alternatives far from common experience. Transitivity assumption may fail due to different framing of the alternatives. For more examples see Mas-Colell, Whinston, Green, et al. (995) Preference relations are described by the means of a utility function which assigns a numerical value to each element in X Def A function u : X Y is a utility function representing the preference relation if for any x, y X x y u(x) u(y) b.) Consider two lotteries L and M. Continuity: If L M N, then there exists a probabilityp [0, ] such that pl + ( p)n M where the notation on the left side refers to a situation in which L is received with probability p andn is received with probability ( p) Independence of irrelevant alternatives: If L M then for any N and p (0, ]: pl + ( p)n pm + ( p)n The independence axiom says that the preference ordering of two lotteries is not changed if each of them is mixed with a third lottery in the same way Problem: The Allias Paradox discussed during the lecture demonstrates the violation of this axiom The four main axioms underpinning Expected Utility Theory are: Completeness, Transitivity, Continuity and Independence of irrelevant alternatives 3

4 c.) The empirical evidence against the Expected Utility Theory is numerous. Some examples are listed below: Allias Paradox Many people do not take up the disaster insurance even though premiums for such policies are often subsidized (Kunreuther (978)) Aversion to Probabilistic Insurance (Kahneman and Tversky (979)) Certainty effect: people overweight outcomes that are considered certain relative to outcomes which a merely probable. Framing Effects: Consider the following situation: a person is about to purchase a stereo for 25 dollars and a calculator for 5. The salesman tells him that the calculator is on sale for 5 dollars less at he other branch of the store located 20 min away. The stereo is the same price. Empirical evidence suggests that the fraction of people who decide to go to the other branch is much higher than in the situation when the calculator is the same price and stereo is soled with the 5 dollars discount. Russian Roulette Puzzle: Compare two versions of Russian Roulette game with a 6 shooter gun: game with one bullet and with four bullets. Empirical evidence suggest that the willingness to pay for removal of one bullet is much higher in the first case then in the second one d.) During the lecture apart from Expected Utility Theory the Rank Dependent Utility and Prospect Theory were introduced. Rank Dependent Utility Generalizes expected utility theory to explain the Allais Paradox. Within the Rank Dependent Utility the probabilities of only unlikely extreme outcomes (as opposed to all unlikely events) are overweighted. Provides no explanation for the framing effects. Prospect theory (Kahneman and Tversky (979)) Explains risk-loving behavior in domain of losses and risk averse in the domain of gains. By introduction of a reference point, explains framing effects. In Prospect Theory the decision making process consists of two phases: editing and subsequent evaluation. In the absence of editing phase the violation of stochastic dominance would be possible. NB The ideas from Rank Dependent Utility Theory were combined with Prospect Theory and laid foundation for the Cumulative Prospect Theory. 3. Insurance Demand under Prospect Theory a.) Loss aversion of the individual implies: v(x) < v( x) x α < λx α λ > 4

5 Diminishing marginal sensitivity implies the concavity of the value function v(x) in the domain of gains: v (x) < 0 for x > 0 α (α )x α 2 < 0 for x > 0 α < and the convexity of value function v(x) in the domain of losses: v (x) > 0 for x < 0 λ α (α ) x α 2 > 0 for x < 0 α < b.) Final wealth, gains and losses with respect to status quo reference point are given in the Tabel 2 State No Loss Loss Probability π π Reference Point W 0 W 0 L Final Wealth W 0 pc W 0 L + ( p)c Gain/Loss pc ( p)c Table 2: Status Quo as a reference point The individual evaluates various alternatives using the following expression: V = πv([ p]c) + ( π)v( pc) The First order conditions on the optimal level of cover are given by: π( p) v (( p)c) ( π) p v ( pc) = 0 p=π v (( p)c) = v ( pc) An interior solution might not exist since in the case of diminishing sensitivity both v (( p)c) and v ( pc) are decreasing in C c.) Note that for status quo reference point that corresponds to the case of no insurance one has V status quo = 0. Thus the insurance won t be purchased if V < V status quo = 0. For a Kahneman Tversky value function and a fair premium rate p = π we have: V = p(( p)c) α λ( p) pc α < 0, for any C > 0 p( p) α λ( p)p α < 0 ( p) α λp α < 0 ( p) α < λ p α p α < λ( p) α p ( + λ α ) < λ α V < 0 p < λ α + λ α Thus the risks that occur with a probability smaller than λ α won t be insured +λ α In the absence of diminishing marginal sensitivity (v (x) = 0 α = ) we have V = p( p)c λ( p)pc = p ( λ)( p)c < 0 C > 0 if λ > The individual won t demand any insurance since in case of loss aversion he dislikes the actuarially fair risks and prefers to stay in uninsured 5

6 d.) Consider the case when the individual s reference point is his final wealth under full insurance (i.e. W = W 2 = W 0 pl). The choice of this reference point could be justified by the fact that some people seem to take one of the alternatives presented (usually a safe option) and evaluate the outcomes of other alternatives with respect to it. Final wealth, gains and losses with respect to full insurance reference point are given in the Tabel 3 State No Loss Loss Probability p p Reference Point W 0 pl W 0 pl Final Wealth W 0 pc W 0 L + ( p)c Gain/Loss p(l C) ( p)(l C) Table 3: Full Insurance as a reference point V = λp ( ( p)(l C)) α + ( p)(p(l C)) α > 0, λ( p) α + p α > 0 p α > ( p + ) < p < λ α λ α λ ( p) α + λ α C < L Using the hint we have that for V > 0 the indifference curve is concave. The situation is illustrated in Figure. Since the indifference curve is concave, the indifference curve that passes through no insurance point will lie above the budget line (straight line between the reference point and no insurance point), thus the optimal cover is given by C = 0 The empirical evidence supports the values α = 0.88, λ = this implies that for risks with probability of occurrence p > the insurance will be purchased and only very rare risks won t be insured. More reference for this task can be found in Schmidt (205) References Christian Gollier. Insurability. Encyclopedia of Actuarial Science, Daniel Kahneman and Amos Tversky. Prospect theory: An analysis of decision under risk. Econometrica: Journal of the Econometric Society, pages , 979. Howard Kunreuther. Disaster insurance protection: Public policy lessons. John Wiley & Sons, 978. Andreu Mas-Colell, Michael Dennis Whinston, Jerry R Green, et al. Microeconomic theory, volume. Oxford university press New York, 995. Ulrich Schmidt. Insurance demand under prospect theory: A graphical analysis. Journal of Risk and Insurance,

7 Figure : Two states of the world diagram for the case when full insurance is treated as a reference point. Axes correspond to deviations from the reference points in state (where the loss ocurs) and 2 (where no loss occurs). Reference point corresponds to x = x 2 = 0 7