Question 1

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 mi minutes before the next person buys one depends on the

current number of mobile phone owners i , but not the time t .

(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 P t ij ( ),

where P t ij ( ) denotes the probability that there will be j mobile phone owners

by time s t + , 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 i

m

A 17-year with-profits endowment assurance is issued to a life aged exactly 48, having a basic guaranteed sum assured of 25,000. The sum assured plus all declared reversionary bonuses to date are paid on survival to the end of the term or immediately on earlier death. Calculate the expected present value of this policy benefit assuming: future bonuses are declared at the rate of 2.5% pa compound, being added in full at the start of each policy year AM92 Select mortality 6.6% pa interest. A woman aged 67 exact takes out an annuity that makes monthly payments in arrears. The first monthly payment is f1,500, and payments increase by 0.23726% each month. Calculate the expected present value of the annuity using the following basis: Mortality: PFA92C20 Interest: 7% per annum [4 A whole life assurance policy pays 20,000 on death in Year 1, 20,100 on death in Year 2, and so on increasing by 100 each year. The payment is made immediately on death of a life currently aged 35 exact. (i) Write down an expression for the present value random variable of this payment, in terms of the curtate future lifetime Kx , and/or the complete future lifetime Tx . [1 (ii) Calculate the expected present value of these benefits, assuming: (a) AM92 Select mortality and 6% pa interest (b) a constant force of mortality of 0.015 pa and force of interest 0.03 pa. [11 [Total 12 A whole life annuity with continuous payments is due to commence in 15 years' time. It will be payable to a life that is currently aged exactly 50, provided that person is still alive when the annuity is due to start. Payments commence at the rate of 20,000 pa, and increase continuously thereafter at a rate of 2% pa compound. Calculate the expected present value of these payments on the following basis: Mortality: PMA92C20 prior to age 65 A constant force of 0.038 pa at ages over 65 Interest: 3% pa effective [6