Answer all the questions below well
An n-state, time-homogeneous Markov jump process with transition probability matrix P(t) over a period of length t, is said to have a stationary distribution if: (1) AP(1) = 1 (2) Osm; $1 for each i = 1, 2, . .., n (3) iEl (i) Show that condition (1) is equivalent to # E =0 where > is the matrix of transition rates. [1] In a particular company the salary scale has only two different levels. On average, an employee spends 2 years at level 1 before moving on to the higher level, or leaving the company. An employee at the maximum level spends an average of 5 years before leaving. Nobody is demoted, and promotion can occur at any time. Upon leaving level 1, the probability that an employee moves to level 2 is 50%. (ii) Explain how you could model this as a Markov process, commenting on any assumptions that you make. [2] (iii) Derive the generator matrix of the Markov jump process. [2] (iv) The company currently has 1,000 employees. The proportions at levels 1 and 2 are 60% and 40% respectively. Assuming that nobody joins the company in the future, determine the distribution of these employees in five years' time. [6] [Total 11 ]A loan of f120,000 is repayable by equal quarterly payments for 25 years. The effective rate of interest is 6% pa. Calculate the interest portion of the first payment. A loan of f1,000 is to be repaid by level monthly instalments over 10 years using an interest rate of 10% pa effective. Calculate the capital repaid in the sixth year. A bank issues a 10-year loan for f100,000 to a businessman. The loan is to be repaid by annual repayments, payable in arrears, calculated using an interest rate of 8% pa effective. The repayment schedule has been designed so that half the capital will have been repaid by the end of the term. The remaining f50,000 will be repaid at the end of the term using funds from other sources. Calculate the annual repayment. A loan of f80,000 is repayable by eight annual payments, with the first payment being made in one year's time. The first three payments are half as much as the remaining five payments, and the annual effective interest rate is 4.5%. Calculate the loan outstanding one year before the loan is completely repaid. A customer borrows f4,000 under a consumer credit loan. Repayments are calculated based on an APR of 15.4%, and are paid monthly in arrears for 5 years. Calculate the amount of each monthly repayment. A man borrows f7,500 to buy a car. He repays the loan by 24 monthly instalments of $368.75, payable in arrears. Calculate the APR on this transaction. [4] A loan of f50,000 is repaid over a period of 10 years by a series of level monthly instalments. Interest is charged on the loan at the rate of interest of 8% pa effective. (i) Calculate the monthly repayment. [2] (ii) Calculate the amount of interest paid in the first year. [3] After the payment at the end of 7 years, the borrower takes a 2-month payment break, ie the borrower does not pay the next 2 monthly instalments. (iii) Calculate the extra amount that needs to be paid each month in order to fully repay the debt by the end of the 10th year. [4] [Total 9]
