Question
A recurrence relation is an equation that recursively defines a sequence of values, whereby each element of a sequence can be written as a function of preceding element(s); the first element of the sequence will be uniquely defined by an initial value of the recurrence relation.
Specifically, if a sequence un can be expressed as a function of only n and the immediate preceding element un−1, i.e., un = g(n, un−1), then we say that un is a recurrence relation of order 1. The values of the entire sequence can be calculated recursively starting from the initial value say u1 and then by u2 = g(2, u1) and more generally un = g(n, un−1) for n = 3, 4, 5, · · · .
(a) Write down an efficient recurrence relation for an . Explain your thought process in words (e.g., using the timeline approach) or prove the result mathematically from first principles. Also write down the initial value for the sequence. (3 marks)
(b) Given an effective discrete periodic rate of 4% per period, tabulate the values of an for n = 1, 2, · · · , 40. (2 marks)
(c) Repeat parts (a) and (b) for (Ia)n instead of an . (5 marks)
(d) Suppose that we would like to develop the relevant calculations for a new actuarial
notation defined as (Qa)n = ∑n t=1 t2vt. Repeat parts (a) and (b) for (Qa)n instead of
an . (7 marks)
(e) Simplify (I ̈a)n − (Ia)n and hence derive a formula for (Ia)n . (3 marks)
(f) Suppose that (Q ̈a)n = ∑n t=1 t2vt−1, simplify (Q ̈a)n − (Qa)n and hence derive a formula for (Qa)n . (4 marks)
(g) Using the formulae developed in parts (e) and (f), re-calculate the relevant values of annuity functions in parts (c) and (d). (2 marks)
(h) Comment on the advantages and disadvantages of the two different approaches (recurrence relation and algebraic formula) used in constructing the relevant tabulated values above. (4 marks)
Suppose that for calculation purpose we only have the three tables of values obtained above, calculate the following by making use of at least one of these three tables:
(i) A loan of $700,000 is to be fully repaid by 25 level annual repayments made in arrears at an effective annual rate of 4% per annum. Calculate the amount of the level annual repayment. (2 marks)
(j) A loan of $450,000 is to be fully repaid by level semi-annual repayments made in arrears for the next 8 years. The equivalent constant force of interest for this loan is 7.8441426% per annum. Calculate the amount of the capital repayment between the 3rd and the 6th year. (5 marks)
(k) Calculate the present value at time 0 of a 10-year continuous annuity with a payment rate of $300 per annum under an effective annual rate of 1.9804%. (4 marks)
(l) Given an effective annual rate of 4%, calculate the present value at time 0 of a 20-year arithmetically increasing annuity immediate whereby the first annual payment is $6,000 and subsequent annual increment is $500. (3 marks)
(m) Consider a 25-year annuity immediate with a payment amount of (26 − t)2 at the end of year t. For instance, the payment amount at the end of year 6 is $400. Calculate its present value at time 0 given an effective annual rate of 4%. (6 marks)
(n) Clarity of report and Excel worksheets, explanation of steps / approaches / rationales taken in solving the questions, communication of findings / results etc. (10 marks
