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13. A consumer who obeys the von Neumann-Morgenstern axioms and has an initial wealth of 160,000 is subject to a fire risk. There is a 5 percent probability of a major fire with a loss of 70,000 and a 5 percent probability of a disastrous fire with a loss of 120,000. Her utility function is U= wo. She is offered an insurance policy with the deductibility provision that she bear the first 7620 of any fire loss. What is the maximum premium that she is willing to pay for this policy?Al (i) Define each of the followimmples of a stochastic process (a) a symmetrkimpleandom walk (b) a compound Poisson process [2] (ii) For each of the processes in (i), classify it as a stochastic process according to its state space and the time that it operates on. [2] [Total4] A2 You have been commissioned to develop aopoject the assets and liabilities of an insurer after one year. This has been requested following a change in the regulatory capital requirenSufficientapitamusnowbe hell such thahere is less than a 0.5% chance of liabilities exceeding assets after one year. The company does not have any existing stochastic models, but estimates have been made in the planning process of "worst casescenarios. Set out the steps ywould take in the development of the model. [6] A3 Let Yj, Yy, Y's, ..., be a sequence of independent and identically distributed random variables with P( Yzkil = 1) = P( V2kil =-1) = =, k=0, 1, 2, .. and defind, = EkHI / Yak- fork= 1, 2, .... (i) Showthat Y, : k=1, 2,.; is a sequence of independent and identically distributed random variables. Hint You may use the fact that, Ifare two variables that take only two values anE( XY) = E( X) E(Y), thenX, Yare independent. [4] (ii) Explain whether or hint k=1, 2,..; constitutes a Markov chain. [1] (iii) (a) State the transition probabilities = P(Ymen = /| Ym= i) of the sequenceY, : k=1,2,...; . (b) Hence show that these probabilotnot depend on the current state and that they satisfy the Chapkindmogorov equations. [3] [Total 8]Al A manufacturer uses a test rigfilmate the failure rate in a batch of electronic components. The rig holds 100 components and is designed to detect when a component fails, at which point it immediately replaces the component with anoth from the same batch. The following aredcoordeh of theomponents used in the testit 1,2,...,n): si= time at which comporiphatced on the rig t = time at which componemoved from rig 1 Component removed due to failure Component working at end of test period The test rig was fully loaded and was run for two yearssontin You should assume that the force of fajbfraecomponent is constant and component failures are independent. (i) Show that the contribution to the likelihood from consiponent exp -u(t - s;)).uf [2] (ii) Derive the maximlikelihood estimatorfor [4] [Total 6] A2 The price of a stock can either take a value above a certain point (state A), or take a value below that point (state B). Assume that the evolution of the stock price in tim can benodelledy a twestat Markov jump process with homogeneous transition rateso AB= 6, BA= P. The process starts in stateta atand time is measured in weeks. (i) Write down the generator matrix of the Markov jump process. [1] (ii) State the distribution of the holding time in each of states A and 19. (iii) If o=3, find the value ofuch that the probability that no transition to state B has occurred until time.2. [2] (iv) Assuming all theformation about the price of the stock is available for a time interval , explain how the model paramearly can be estimated from the available data. [2] (v) State what you would testetermine whether the data support the assumption of a tstate Markov jump process model for the stockprice. [Total 7]