Math 134 Tutorial 7, 2011: Financial Maths

Transcription

1 Math 134 Tutorial 7, 2011: Financial Maths For each question, identify which of the formulae a to g applies. what you are asked to find, and what information you have been given. Final answers can be worked out at home or during free periods. Formulae used in Financial Maths: a Amount of Simple Interest I = P in b Future Value using Simple Interest: S = P in c Future Value using Compound Interest: S = P i n d Effective Rate of Interest: i eff = i m m 1 where m is the no. of compounding periods in 1 year. [ i n ] 1 e Future Value of an Annuity: S = Rs n i = R i [ ] 1 i n f Present Value of an Annuity: A = Ra n i = R i [ ] 1 i n g Present Value of an Annuity paid in Advance: A = Ra n i i = R i i h Total Interest = Difference between the Sum of Payments and the Present or Future Value. 1. Shona s granny gives her a loan of R to be repaid after 6 years at 8% annual simple interest. How much will Shona have to pay 6 years from now to settle the loan? 2. How much would Shona have to pay if interest was compounded a annually? b daily? 3. What is the annual simple interest rate that will make R400 grow to R450 in a 2 years? b 10 months? 4. Calculate the effective rate of interest for a 16.7% p.a. compounded monthly b 16.1% p.a. compounded daily c 17% p.a. compounded semi-annually d 16.9% p.a compounded quarterly e 17.2% p.a. compounded annually. 5. A fixed deposit is known to have an effective rate of 8.5%. If interest is compounded monthly, find the nominal rate of interest. 6. Your client just turned 35 and will retire at the end of the month she turns 55 years old. She starts saving R1 000 at the end of each month, earning 7.2% p.a. compounded monthly. How much will she have in her account when she retires? 7. Find the present value of R5 500 due in 7 years from now at an interest rate of 9% p.a. compounded semi-annually. 8. Londiwe amortizes a home loan of R by obtaining a 20-year mortgage at the rate of 9.5% p.a. compounded monthly. Find a the monthly payment b the total interest paid c the principal remaining after 5 years 9. What nominal interest rate, compounded quarterly, would be needed for R1400 to grow to R2000 in 4 years? 1

2 10. What nominal interest rate, compounded daily, would be needed for R1400 to grow to R2000 in 4 years? 11. Yaseen bought a computer for R5 200 and agreed to pay it off by making monthly payments of R200. If the shop charges interest at a rate of % p.a. compounded monthly, how many months would it take to pay off the debt?. How many days will it take for an investment of R450 to grow to R600 at 18% p.a. compounded daily? 13. How long will it take an investment to double if interest is 15% p.a. compounded monthly? 14. Andile is the beneficiary of a trust fund set up at his birth 21 years ago. Originally R was invested in the trust fund. What will Andile receive from the trust fund if the money has earned interest at 7% p.a. compounded a annually b monthly c daily? 15. Determine the finance charge i.e. the interest paid on a 36-month R0 000 car loan with monthly payments if interest is charged at 11.5% p.a. compounded monthly. 16. A computer was bought for R 000 on 23 March What was the annual rate of depreciation on reducing or diminishing balance if the computer is worth R4 000 on 23 March 2010? 17. Suppose you want to have R1 million saved when you retire at age 60. How much should you save at the end of each month if interest is 8% p.a. compounded monthly and you start saving when you are a 40 years old i.e. for 20 years? b 35 years old? c 25 years old? 18. Suppose you have the choice of taking out a R loan at 10% p.a. compounded monthly for either 30 years or 15 years. How much savings is there in the finance charge if you choose the 15-year loan? 19. Simon can get married when he has saved R How much should he save at the end of each month at 9% p.a. compounded monthly if he want to get married a 4 years from now? b 2 years from now? 20. Simon calculates that he can only save R250 per month for his wedding. How long will he have to save at 9% p.a. compunded monthly before he has accumulated R50 000? 21. A loan of R is being amortized over 25 years at an interest rate of % p.a. compounded monthly. Find a the monthly payment b the outstanding balance at the beginning of the 101st month. c the interest paid in the 101st month d the total interest paid in 25 years. ANSWERS 1. b. Find S if P = , n = 6, i = 0.08; R a c. Find S if n = 6, i = 0.08; R b c. Find S if n = 2190, i = 0.08 ; R a b. Find i if S = 450, P = 400, n = 2; 6.25% b b. Find i if S = 450, P = 400, n = 10 ; 15% 2

3 4. d or c a 18.04% b 17.46% c 17.72% d 18.00% e 17.2% 5. d. Find i if i eff = 0.085, m = ; 8.19% 6. e. Find S if n = 240, i = 0.072, R = 1000; R c. Find P if S = 5 500, n = 7 2 = 14, i = 2 ; R a f. Find R if A = , i = 5, n = 20 = 240; R b h = R c f. Find A if i = 5, n = 15 = 180, R = ; R c. Find i if S = 2000, P = 1400, n = 16 and i is replaced with i 4 ; 9.02% 10. c. Find i if S = 2000, P = 1400, n = 4 = 1460 and i is replaced with i ; 8.92% 11. f. Find n if A = 5200, R = 200, i = 0.. c. Find n if S = 600, P = 450, i = 0.18 ; days = 0.01; months 13. c. Find n if S = 2P, P = P, or S = 2, P = 1, or S = 200, P etc. i = 0.15 ; months 4 years 8 months 14. a c. Find S if P = , n = 21, i = 0.07; R b c. Find S if P = , n = 21 = 252, i = 0.07 ; R c c. Find S if P = , n = 21 = 252, i = 0.07 ; R f and h. Find R if n = 36, A = 0 000, i = ; R = Interest = R c. Find i if P = 000, S = 4 000, n = 5; decrease of 19.73% per year. 17. a e. Find R if S = , i = 0.08, n = 240; R b e. Find R if S = , i = 0.08, n = 300; R c e. Find R if S = , i = 0.08, n = 420; R f and h. Find R if A = , i = 0.1, n = 360; R = Interest = R Find R if A = , i = 0.1, n = 180; R = Interest = R The savings from paying off in 15 years is = R a e. Find R if S = , i =, n = 48; R b e. Find R if S = , i =, n = 24; R e. Find n if R = 250, i =, S = ;.10 years 3 months 21. a f. Find R if A = , i = 0. b f. Find A if R = , i = 0. = 0.01, n = 25 = 300; R = 0.01, n = 200; R c a. Find I if P = , i = 0.01, n = 1; R d h. n = 300, R = ; R

4 Math 134 Tutorial 7, 2011: Financial Maths Solutions Tutors Version with questions and answers 1. Shona s granny gives her a loan of R to be repaid after 6 years at 8% annual simple interest. How much will Shona have to pay 6 years from now to settle the loan? S = P in = = How much would Shona have to pay if interest was compounded a annually? b daily? a S = P i n = = b S = P i n = = What is the annual simple interest rate that will make R400 grow to R450 in a 2 years? b 10 months? a S = P in where S = 450, P = 400, n = = 400 2i i = = = 6.25% b S = P in where S = 450, P = 400, n = = i i = = 0.15 = 15% 4. Calculate the effective rate of interest for a 16.7% p.a. compounded monthly i eff = i n 1 = = = 18.04% b 16.1% p.a. compounded daily i eff = i n 1 = = = 17.46% c 17% p.a. compounded semi-annually i eff = i n 1 = = = 17.72% d 16.9% p.a compounded quarterly i eff = i n 1 = = = 18.00% e 17.2% p.a. compounded annually. 17.2% no calculation needed - the effective rate is the percentage by which money actually grows in a year 5. A fixed deposit is known to have an effective rate of 8.5%. If interest is compounded monthly, find the nominal rate of interest. i eff = i n = x = 8.19% 6. Your client just turned 35 and plans to retire on her 55th birthday. At the end of each month she saves R1 000, earning 7.2% p.a. compounded monthly. How much will she have in her account when she retires? R = 1000, i = 0.072, n = 20 = 240 i n S = Rs n i = R = 1000 = i Find the present value of R5 500 due in 7 years from now at an interest rate of 9% p.a. compounded semi-annually. S = 5500, n = 7 2 = 14 semi-years, i = 2 = S = P i n 5500 = P P = =R

5 8. Londiwe amortizes a home loan of R by obtaining a 20-year mortgage at the rate of 9.5% p.a. compounded monthly. Find a the monthly payment b the total interest paid c the principal remaining after 5 years a A = , i = 5, n = 20 = = R R = b Total amount paid = = Loan amount = The interest is the difference between what is paid and what the loan amount was = c Find the present value of the remaining 15 years of payments after 5 years of payments have passed A = = What nominal interest rate, compounded quarterly, would be needed for R1400 to grow to R2000 in 4 years? P = 1400, S = 2000, n = 4 4 = 16 S = P i n 2000 = i 16 Divide by 1400 first. i = = 017 = 9.02% 10. What nominal interest rate, compounded daily, would be needed for R1400 to grow to R2000 in 4 years? P = 1400, S = 2000, n = 4 = 1460 S = P i n 2000 = 1400 i 1460 Divide by 1400 first. i = = = 8.92% 11. Yaseen bought a computer for R5 200 and agreed to pay it off by making monthly payments of R200. If the shop charges interest at a rate of % p.a. compounded monthly, how many months would it take to pay off the debt? A = 5200, R = 200, i = 0. = = n 0.01 Before taking logs, isolate 1.01 n 0.26 = n log 0.74 = log 1.01 n log 0.74 n = log 1.01 n = months In reality, there will either be 31 payments with the last payment smaller, or the 30th payment will be increased to reduce the balance of the debt to 0. 5

6 . How many days will it take for an investment of R450 to grow to R600 at 18% p.a. compounded daily? P = 450, S = 600, i = 0.18 S = P i n 600 = n = log log 0.18 = days n 13. How long will it take an investment to double if interest is 15% p.a. compounded monthly? P = 100 or any value, or just use P, S = 200 or S = 2P, i = 0.15 S = P i n 200 = n 2 = 0.15 n log 2 n = log 0.15 = months 4 years 8 months 14. Andile is the beneficiary of a trust fund set up at his birth 21 years ago. Originally R was invested in the trust fund. What will Andile receive from the trust fund if the money has earned interest at 7% p.a. compounded a annually b monthly c daily? a S = P i n S = = b S = P i n S = = c S = P i n S = = Determine the finance charge i.e. the interest paid on a 36-month R0 000 car loan with monthly payments if interest is charged at 11.5% p.a. compounded monthly. First find the payments: A = 0000, n = 36, i = = R R = Total amount paid = = Subtract the amount loaned: Finance charge = Without rounding the answer is , which is slightly more accurate. 16. A computer was bought for R 000 on 23 March What was the annual rate of depreciation on reducing or diminishing balance if the computer is worth R4 000 on 23 March 2010? P = 000, S = 4000, n = 5 S = P 1 i n 4000 = i = 1 i i = = = 19.73% 17. Suppose you want to have R1 million saved when you retire at age 60. How much should you save at the end of each month if interest is 8% p.a. compounded monthly and you start saving when you are a 40 years old i.e. for 20 years? S = , i = 0.08, n = 20 = 240 payments = R 0.08 R = =

7 b 35 years old i.e. for 25 years? S = , i = 0.08, n = 25 = 300 payments = R R = c 25 years old i.e. for 35 years? S = , i = 0.08, n = 35 = 420 payments = R R = Suppose you have the choice of taking out a R loan at 10% p.a. compounded monthly for either 30 years or 15 years. How much savings is there in the finance charge if you choose the 15-year loan? Find the payments for each case, then work out the interest paid in each case and subtract one from the other. For 30 years: A = , i = 0.1, n = = R R = Total amount paid = = Interest paid = Total amount paid - loan amount = For 15 years: A = , i = 0.1, n = = R R = Total amount paid = = Interest paid = Total amount paid - loan amount = The savings from paying off in 15 years is Miguel can get married when he has saved R How much should he save at the end of each month at 9% p.a. compounded monthly if he want to get married a 4 years from now? S = , i =, n = 4 = 48 payments = R R = b 2 years from now? S = , i =, n = 2 = 24 payments = R R =

8 20. Miguel calculates that he can only save R250 per month for his wedding. How long will he have to save at 9% p.a. compunded monthly before he has accumulated R50 000? S = , i =, R = 250 n = = n = n = n Now take logs or write in log form log = n log log n = log = 2.6 months = 10 years, 3 months 21. A loan of R is being amortized over 25 years at an interest rate of % p.a. compounded monthly. Find a the monthly payment A = , i = 0. = 0.01, n = 25 = 300 payments = R R = b the outstanding balance at the beginning of the 101st month. This is the present value of the 200 payments that still need to be paid after 100 payments have been made A = = c the interest paid in the 101st month = d the total interest paid in 25 years. This is the difference between the total amount paid 300 payments of R and the loan amount =