Exam Introduction Mathematical Finance and Insurance

Transcription

1 Exam Introduction Mathematical Finance and Insurance Date: January 8, Duration: 3 hours. This is a closed-book exam. The exam does not use scrap cards. Simple calculators are allowed. The questions may be answered in English or in Dutch. This exam has 5 pages. 1. After the exam period, you will go on a holiday to the US. You are looking to buy a travel insurance policy. The risk X you face has the following distribution: x P [X = x] Your utility is driven by power utility with parameter 2/3, u(w) = w 2/3. The travel insurance company has an exponential utility function with parameter α = The initial wealth equals w 0 = and W 0, respectively. a. When you run an insurance company you face several risks. What are the main risks used in the supervision of insurance companies (Solvency II)? b. Under the assumptions described above and using the equilibrium equation from using expected utility, determine the minimum premium P the insurer wants to receive. c. Under the assumptions described above and using the equilibrium equation from using expected utility, what would be the maximum premium P + you are willing to pay to the insurer? Does this result in a transaction? d. On the basis of the formulas used in the derivation in b., show that P can be approximated by α. Do we have a 1

2 transaction when using this approximation? What do you learn from this? e. The travel insurer is considering to buy reinsurance where the reinsurance premium is based on the expected value premium principle with safety loading λ. Would you recommend a proportional or a stop-loss reinsurance contract? Why? 2. A bonus-malus scale with n steps is often used in car insurance. In the case of no claims during the past year, a driver will go up one step on the scale. In the case of claims during the past year, the driver will go down the scale with at least one step. The bonus-malus scale can be modeled using Markov Chains. The state space is 1,..., n and X t denotes the (random) state at time t for t = 0, 1,.... Assume that the sequence of random variables X 0, X 1, X 2,... is a stationary, discrete time Markov Chain. Also assume that the random variables C t = number of claims for a driver in year t for t = 1, 2,... are i.i.d. Furthermore, denote the number of rides in each year by m, constant through time. On each ride, the probability of a claim equals p. a. If we assume that the rides are independent, what would be a suitable distribution for modeling C t? Furthermore, we assume that n = 2 and the bonus malus scale is as follows: New step after... claims Step Premium b. Express the transition matrix P in terms of m and p. c. Compute the steady state distribution in terms of m and p. d. The steady state premium b(m, p) is a function of m and p. Determine b(m, p). e. For b(m, p) it holds that for a given m, b(m, p) is increasing in p. Similarly, for a given p, b(m, p) is increasing in m. Please explain. Note: you do not need to prove these relationships. 3. Suppose you run an insurance company and you face a non-negative risk X. To try to avoid a bankruptcy, you are risk averse. When 2

3 considering the premium you charge for the risk X, several premium principles are available. We focus on the variance premium principle, so π[x] = E[X] + αvar[x] with α > 0. a. Define the following four desirable properties: non-negative loading, no rip-off, consistency and additivity. b. Check whether the variance premium principle satisfies the no rip-off property for all values of α. If so, prove it. If not, provide a counterexample or disprove it. After meeting with several other insurers, you decide to underwrite policies by a pool of insurers (co-insurance). Losses are shared according to a predetermined ratio. The total risk is denoted by S and each insurer i = 1,..., n accepts a proportion r i of the risk, S i = r i S, where 0 r i 1 and r i = 1. Each insurer employs the variance premium principle with parameter α i, i = 1,..., n. c. Show that minimizing the total premium results in proportions r i that are inversely proportional to α i. Interpret. 4. The equations below describe competitive equilibrium in a singleperiod market with uncertainty. The notation is as follows: k is the number of agents, u i (x) (i = 1,..., k) are their utility functions, n is the number of possible future states, and p j (j = 1,..., n) are the objective probabilities of these states. Moreover, the initial payoff of agent i in state j before trading is given by ω j i (i = 1,..., k; j = 1,..., n), whereas the payoffs in equilibrium are denoted by x j i (i = 1,..., k; j = 1,..., n). The utility functions satisfy the standard properties u i(x) > 0 and u i (x) < 0 for all x. q j x j i = q j ω j i (i = 1,..., k) (1) q j u i(x j i ) = λ i p j (i = 1,..., k; j = 1,..., n) (2) k k x j i = ω j i (j = 1,..., n) (3) q j = 1 (4) a. Explain the reasons why these equations are imposed. Also explain the meaning of the variables that appear in the equations and that were 3

4 not already mentioned above. [8 pts.] b. Show the following property: if there are two states j 1 and j 2 such that k k ω j 1 i = ω j 2 i then the equilibrium allocation satisfies x j 1 i = x j 2 i for all i = 1,..., k. [8 pts.] c. Describe the meaning of the property in part b. in words. 5. Let a single-period market with two possible future states and two assets be given as follows: current price in price in price up state down state S 0 S u S d B 0 B u B d Assume that the following inequalities hold: S u S 0 > B u B 0 > B d B 0 > S d S 0. Prove that this market does not allow arbitrage. [8 pts.] 6. Consider the following tree model for the price of a risky asset: Assume that the discretely compounded riskfree interest rate is 2% per period. One period corresponds to one step in the tree. a. Determine the price of a European put option with strike K = 60. b. Determine the price of an American put option with the same strike. 4

5 7. Consider a market that is given by a geometric tree model with parameters u, d, and r. In other words, there are two assets called S and B which have initial values S 0 and B 0 respectively and whose prices evolve according to the rules S γu = us γ, S γd = ds γ, B γu = B γd = (1 + r)b γ where γ denotes a general node in the tree. It is assumed that u > 1 + r > d > 0. a. Prove that the implied probability q of an up move is the same for every node in the tree. b. Let p be a given number such that 0 < p < 1. Prove that it is possible, given an initial capital V 0, to define a trading strategy which satisfies the budget constraint at every node and which is such that the portfolio value at a node γ that is reached after j up moves and k j down moves is given by V γ = pj (1 p) k j q j (1 q) k j (1 + r)k V 0. Derive explicit expressions for the amounts of units of both assets that are to be held in the portfolio at each node, and prove that the percentage of portfolio value held in risky assets (S) is the same at each node. [10 pts.] c. Prove that the portfolio strategy defined above provides an optimal solution for the maximization problem maximize E P [ ln(v N )] subject to budget constraint at each node, initial portfolio value = V 0 where N > 0 is a given integer, and the symbol E P indicates that expectation is taken under the assumption that the objective probability of an up move at each node is given by p. 5