PREMIUM VERSION PREVIEW

Transcription

1 FINANCIAL MATHS

2 PREMIUM VERSION PREVIEW

3 205 LCHL Paper Question 6 (a) (i) Donagh is arranging a loan and is examining two different repayment options. Bank A will charge him a monthly interest rate of 0.35%. Find, correct to three significant figures, the annual percentage rate (APR) that is equivalent to a monthly interest rate of 0.35%. 0 Marks 4 for correct substitution into formula. Find out what 00 would grow to in 2 months at 0.35% per month. Compound Interest Formula F = P + i t F =? P = 00 i = t = 2 F = P + i t F = F = Interest earned i = %

4 205 LCHL Paper Question 6 (a) (ii) Bank B will charge him a rate that is equivalent to an APR of 4.5%. Find, correct to three significant figures, the monthly interest rate that is equivalent to an APR of 4.5%. 0 Marks 4 for correct substitution into formula. Compound Interest Formula F = P + i t We do not know what P and F are so we can choose any initial amount, P. The easiest is to let P =. By doing that we can say that after year P will have earned 4.5% interest so F = There are 2 months in a year so t = 2 and we need to solve for i..045 = + i = + i = i i = i = 0.367%

5 205 LCHL Paper Question 6 (b) Donagh borrowed at a monthly interest rate of 0.35%, fixed for the term of the loan, from Bank A. The loan is to be repaid in equal monthly repayments over ten years. The first repayment is due one month after the loan is issued. Calculate, correct to the nearest euro, the amount of each monthly repayment. The sum of the present value of loan repayments A is equal to the amount of the loan. A A A A = 80,000 Sum of a Geometric Series a rn S n = r A a =.0035 r =.0035 n = 20 A A = 80,000 A = 80, A = A = 80,000 Alternate Method: Amortisation Formula Amortisation Formula A = P i + i t + i t P = A =? i = t = 20 i + i t A = P + i t A = 80, A = A 88 5 Marks

6 204 LCHL Sample Paper Question 8 (a) Pádraig is 25 years old and is planning for his pension. He intends to retire in forty years time, when he is 65. First, he calculates how much he wants to have in his pension fund when he retires. Then, he calculates how much he needs to invest in order to achieve this. He assumes that, in the long run, money can be invested at an inflation-adjusted annual rate of 3%. Your answers throughout this question should therefore be based on a 3% annual growth rate. Write down the present value of a future payment of 20,000 in one years time. Present Value P = F + i t = F = P =? i = 0.03 t = (b) Write down, in terms of t, the present value of a future payment of 20,000 in t years time t

7 204 LCHL Sample Paper Question 8 (c) Pádraig wants to have a fund that could, from the date of his retirement, give him a payment of 20,000 at the start of each year for 25 years. Show how to use the sum of a geometric series to calculate the value on the date of retirement of the fund required. Sort of like a loan question, he needs an amount in the bank today that will give him 25 years worth of 20,000 payments. The net present value of the payments he receives needs to be equal to the amount he starts with in the fund Sum of a Geometric Series a rn S n = r a = r =.03 n = S 25 =.03 = 358,

8 204 LCHL Sample Paper Question 8 (d) (i) Pádraig plans to invest a fixed amount of money every month in order to generate the fund calculated in part (c). His retirement is 40 2 = 480 months away. Find, correct to four significant figures, the rate of interest per month that would, if paid and compounded monthly, be equivalent to an effective annual rate of 3%. Compound Interest Formula F = P + i t We do not know what P and F are so we can choose any initial amount, P. The easiest is to let P =. By doing that we can say that after year P will have earned 3% interest so F =. 03. There are 2 months in a year so t = 2 and we need to solve for i..03 = + i = + i 2.03 = i i = i = %

9 204 LCHL Sample Paper Question 8 (d) (ii) Write down, in terms of n and P, the value on the retirement date of a payment of P made n months before the retirement date. Compound Interest Formula F = P + i t F = P + i t F = P n F =? P = P i = t = n

10 204 LCHL Sample Paper Question 8 (d) (iii) If Pádraig makes 480 equal monthly payments of P from now until his retirement, what value of P will give the fund he requires? Each payment stays in the bank for one month less than the previous earning one month less interest. P P P P Sum of a Geometric Series a rn S n = r a = P r = n = 480 P P = P = P = P = =

11 204 LCHL Sample Paper Question 8 (e) If Pádraig waits for ten years before starting his pension investments, how much will he then have to pay each month in order to generate the same pension fund? If he waits 0 years he will only get to deposit = 360 payments. We can see he will have to save a lot more each month to generate the same pension fund. Sum of a Geometric Series a rn S n = r a = P r = n = 360 P P = P = P = = 358,70.84

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