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In a small country there are only four authorised insurance companies. A. E. C and D. The law in this country requires homeowners to take out buildings insurance from an authorised insurance company. All policies provide cover for a period of one year. Based on analysis of the compliance database used to check that every home is insured. the probabilities of a homeowner transferring between the four companies at the end of each year are considered to be described by the following transition matrix: A il ill ill Ill] 3 ill {LEI ll] ill C Ill} ill Ill-I'l {Li D D {12 {12 t? Yolanda has just bought her policy from Company C for the rst time. {i} Calculate the probability that Yuma 1will be covered by Company C for at least ve yeasts before she changes provider. [2] anhary too]: out a policy with Company at in January 1013. Unfortunately. Zaehaiy's house burnt down on 11 March El. {ii} Cale ulate me probability that Company A dam NUT cover anhary's borne at the time of the re. [2] {iii} Calculate the expected proportions ofhomeowners who are covered by each insurance company in the long run. [4] Company .4! makes an offer to buy Company D. It b its purchase price on the assumption that homeowners who would previously have purchased policies from Company A or Company B would now buy from the combined company. to be called Addda. {iv} Determine the transition matrix. which will apply after the takeover if Company A's assumption about homeowners' behaviour is correct [2] {v} Comment on. the appropriateness of Company A's assumption. [1] [Total 12] An energy provider is worried about the number of its customers who transfer to other companies within the first two years of their contract and is trying to direct its advertising towards the most loyal section of the population. The company has looked at its records over recent years and has fitted a Cox proportional hazards model to those who have transferred within the first two years using the factors which appear to have the most impact on early transfer rates. The following figures have been derived from the data: Factor Parameter Variance Estimate Gender Male -0.25 0.015 Female 0 0 Volume of energy High 0.32 0.008 consumed Low 0 0 Area of Residence City Centre 0.19 0.012 City (not centre) Rural -0.35 0.005 (i) Give the hazard function for this Cox proportional hazard model defining all the terms and covariates. [3] (ii) State the features of the person to whom the baseline hazard applies. [1] (iii) Calculate symmetric 95% confidence intervals for the parameters based on the standard errors. [2] (iv) Test the suggestion that women change energy providers more frequently than men. [3] There is a 70% probability that a male customer who is a low consumer of energy and lives in a rural area has transferred providers before the end of two years. (v) Calculate the probability that a male customer who is a high consumer of energy and lives in a city centre remains with the company for at least two years. [3] (vi) Set out how you would determine whether the effect of any of the factors depends upon any of the other factors. [Total 17]