Answer these questions appropriately. Ensure that you respond to all questions.
A mortality investigation is being conducted between the dates 1 January 2004 and 31 December 2007. The data from four lives under consideration was as follows: Life Date of Birth Date of Entry Date of Exit Mode of Exit Pele 11.11.73 24.3.02 29.12.07 Death Johan 1.9.80 30.8.05 Did not leave Gary 10.2.79 10.10.03 21.6.04 Surrender Diego 8.2.82 10.8.05 Did not leave (i) Assuming that the day of entry, but not the day of exit, counts in the exposed to risk, calculate the number of days of exposure contributed to the central exposed to risk by each life at each age. [12] (ii) State what modifications, if any, you would need to make if you were determining the initial exposed to risk and not the central exposed to risk. [2] [Total 14]You are trying to model the number of people who own mobile phones as a Markov jump process. You assume that once someone has bought a mobile phone they will never give it up, but they can only ever own one mobile phone. You also assume that the number of potential phone owners is infinite and every individual will live forever. Finally, you assume that only one individual can buy a phone at any instant, and that the average length of time /; minutes before the next person buys one depends on the current number of mobile phone owners i, but not the time f. (i) Write down the generator matrix (transition rate matrix) for this process. [1] (ii) Write down the integral form of the Kolmogorov forward equations for Pi (t), where Py (t) denotes the probability that there will be / mobile phone owners by time s +f, given that there were i at time s. [3] (iii) Obtain an expression for the probability that exactly one mobile phone will be sold in the next minute if there are currently i people with mobile phones. [2] (iv) After fitting the model to recent data it looks like /, = (0.9999991705). Calculate the expected time before the whole population has a mobile phone if 1,000,000 do at present. Comment on your answer. [4] [Total 10]18 Agency theory suggests that there is a great deal of conflict between the interests of various stakeholders. This has led to a set of observable and significant costs associated with monitoring and protecting individual interests. Arguably, all stakeholders would benefit if they could agree to work together in the best interests of the entity, without incurring these agency-based costs. Explain why it would be difficult, if not impossible, to eliminate these agency costs. [5] 19 An investor has a policy of investing in a diversified portfolio. He has a wide range of investments in stocks and shares, all of which have low Beta coefficients. He is concerned that his returns have not been as healthy as he would have liked over the past several years and he has asked your advice on a number of issues. The return on his portfolio has generally been steady, but it has not been very much better than he might have obtained from certain bank deposit accounts. His return suffered a major setback in 2004 in the aftermath of the tsunami in the Indian Ocean. Investments in companies based in countries affected by the disaster and those linked to the travel industry were badly affected by the crisis. He has been offered the opportunity to liquidate his portfolio and invest in a new overseas property development in Bulgaria. The project has a very attractive forecast return and the Beta coefficient is close to zero. (i) Explain why the return on this investor's portfolio is unlikely to be particularly high. [6] (ii) Explain why his portfolio might have taken a major setback in the aftermath of the tsunami, despite the extent of the diversification. [6] (iii) Explain whether it would be logical to reinvest the value of his portfolio in the Bulgarian development. State the assumptions that you have made in giving your advice. [8] [Total 20]
