MULTIPLE CHOICE 1 Using the recurrence relation t0 = 8, tn+1 = tn + 9, the sixth term would be: A 2 15 B 17 C 23 D 53 E 62 A canoe costs $40 to hire for the first day and $30 for each extra day. If Cn is the cost of hiring the canoe for n days, the recurrence relation is: 3 4 A C0 = 40, Cn+1 = Cn + 30 B C1 = 40, Cn+1 = Cn + 30 C C0 = 40, Cn = Cn+1 + 30 D C1 = 30, Cn+1 = Cn + 40 E C0 = 30, Cn = Cn+1 + 30 The recurrence relation for the sequence 9, 27, 81, ... is: A t0 = 9, tn+1 = 9tn B t0 = 9, tn+1 = tn + 9 C t0 = 9, tn+1 = 3tn D t0 = 9, tn+1 = tn + 18 E t0 = 9, tn+1 = tn + 3 A 7% increase is made by using a common ratio of: A 5 0.07 B 1.07 C 1.7 D 7 E 107 A colony of frogs increases by 20% each year. If there were originally 500 frogs in the colony, the recurrence relation for the number of frogs Fn after n years is: 6 A F1 = 500, Fn+1 = 1.20Fn B F1 = 500, Fn+1 = 0.20Fn C F0 = 500, Fn+1 = 0.20Fn D F0 = 500, Fn+1 = 1.20Fn E F1 = 500, Fn = 1.20Fn+1 The following recurrence relation can be used to model a compound interest investment of $2000, paying interest at the rate of 9% per annum. V0 = 2000, Vn+1 = 1.09Vn After how many years will the value of the investment first exceed $5000? A 7 10 B 11 C 12 D 15 E 17 The first three terms in a sequence using the recurrence relation t1 = 3, tn+1 = 2tn - 5, are: A -3, -1, 1, ... B 1, -3, -11, .. C 2, -5, 3, ... D 3, 1, -3, ... E 3, 2, -5, ... Page 1 of 10 8 In the sequence of patterns below pattern 1 used eight sticks. Pattern 1 Pattern 2 Pattern 3 The number of sticks required to make Pattern 100 would be: A 9 503 B 508 C 795 D 800 E 805 D -0.91 E 0.91 A 9% decrease is made by using a common ratio of: A -109 B -9 C -1.09 10 If a recursive equation t0 = 10, tn+1 = 1.5tn + b generates the sequence 10, 12, 15, 19.5, ..., b is: A -5 B -3 C -2 D 3 E5 11 A recursive equation is T(0) = 3, T(n+1) = T(n) + 4. A rule for the nth term is given by A T(n) = 4n + 3 B T(n) = 4n - 1 C T(n) = n + 2 D T(n) = n + 3 E T(n) = n + 4 The following information relates to Questions 12 and 13. Conservation authorities want to control the water buffalo population in a national park. Each year there is an increase of 10% in the buffalo numbers due to natural factors and, at the end of the year, 150 buffalo are removed by park officials. The size of the buffalo population at the start of 2001 was 1 500. 12 How many buffalo would be in the park at the start of 2003? A 0 B 900 C 930 D 1469 13 A recursive equation that could be used to model this situation is: A t0 = 1 500, tn + 1 = 1.1 tn - 150 B t0 = 1 500, tn + 1 = 1.1(tn - 150) C t0 = 1 500, tn + 1 = 0.1tn - 150 D t0 = 1 500tn + 1 = 0.1(tn - 150) E t0 = 5 000tn + 1 = 0.9tn - 150 Page 2 of 10 E 1500 14 Each trading day, a share trader buys and sells shares according to the rule Tn+1 = 0.6 Tn + 50 000 Where Tn is the number of shares the trader owns at the start of the nth trading day. From this rule, it can be concluded that each day A the trader sells 60% of the shares she owned at the start of the day and then buys another 50 000 shares. B the trader sells 40% of the shares she owned at the start of the day and then buys another 50 000 shares. C the trader sells 50 000 of the shares that she owned at the start of the day. D the trader sells 60% of the 50 000 shares that she owned at the start of the day. E the trader sells 40% of the 50 000 shares that she owned at the start of the day. 15 A sequence is defined by the recursive equation tn+1 = -4tn - 1. If the third term of the sequence is 51, the first term is: A -205 B -13 C 13 D 5 E3 16 At the start of each year, a cosmetics company increases the number of different eye shadow colours it manufactures by 10%. The graph that most closely reflects the pattern of growth is: E 17 An amount of $5000 is invested, earning compound interest at the rate of 4.5% per annum compounding monthly. The effect interest rate is closest to: A 4.3% B 4.4% C 4.5% D 4.6% E 4.7% 18 The interest rate on a compound interest loan is 10.2% per annum, compounding quarterly. The value of the loan after n months, Vn, is modelled by the recurrence relation V0 = 500, Vn+1 = R X Vn. The value of R in this model is: A 1.00 B 0.898 C 1.102 D 1.0255 Page 3 of 10 E 2.55 19 Inga invests $8500 at 6% per annum, compounding quarterly. The amount of interest she earns during the fourth year of the investment is: A $521.59 B $623.62 C $2231.05 D $2286.38 E $4489.32 20 Gregor invests $10 000 and earns interest at a rate of 6% per annum compounding quarterly. Every quarter, after interest has been added, he withdraws $500. At the end of four years, after interest has been added and he has made the $500 withdrawal, the value of the remaining investment will be closest to: A $3 720 B $4 220 C $5 440 D $21 660 E $22 160 (20 marks) SHORT ANSWER PROBLEM ONE Jeremy is planning to save money for his \"Schoolies\" activity at the end of the year. He has an account with $1500 in it already, and each month he will add another $75 from his part-time job. The account pays 4.8 % p.a. interest, compounded monthly. (a) What is the interest rate per month? (1 mark) 4.8/12 = 0.4% (b) Using Bn to represent the balance of the account after n months, write down a recurrence relation to model this investment situation. (2 marks) B0=1500, (c) Bn+1=1.004xBn+75 Use your calculator to determine recursively the value of the investment after Jeremy has made six payments. (2 marks) End of month Calculation details 0 (d) Value of account 1500.00 1 1.004x1500.00+75 1581.00 2 1.004x1581.00+75 1662.32 3 1.004x11662.32+75 1743.97 4 1.004x1743.97+75 1825.95 5 1.004x1825.95+75 1908.25 6 1.004x1908.25+75 1990.89 Jeremy will close the account after twelve (12) months. How much money, correct to the nearest cent, will he have for his \"Schoolies\" activities? $2493.67 Show details of any Finance Solver calculations in the table provided below. (1 mark) N I(%) PV Pmt FV PpY, CpY 12 4.8 -1500 -75 2493.67170 26 12 Page 4 of 10 PROBLEM TWO Jeremy's Aunt Alicia is about to retire and is wondering what to do with her superannuation money. One option is to invest her $625 000 in an annuity paying 4.5 % per annum compounded monthly. (a) How much would she receive per month if the annuity is for 20 years? $3954.06 Show details of any Finance Solver calculations in the table provided below. (1 mark) N I(%) PV Pmt FV PpY, CpY 240 4.5 -625000 3954.05860 13 0 12 (b) If she decided that she wanted to receive $4000 per month, how many fewer payments would she receive from the annuity? 236 payments , so 4 fewer payments, if last payment was $2283.20 to make FV = 0 Show details of any Finance Solver calculations in the table provided below. (c) (2 marks) N I(%) PV Pmt FV PpY, CpY 235.5708 4.5 -625000 4000 0 12 Another annuity company claims that their annuity product would give her $4250 per month for twenty years for the $625 000 investment. What interest rate, correct to two decimal places, would give this return? 5.36% p.a. Show details of any Finance Solver calculations in the table provided below. (1 mark) N I(%) PV Pmt FV PpY, CpY 240 5.359925 -625000 4250 0 12 Alicia believes that either twenty year annuity discussed above will not last long enough for her, so she looked at investing in a perpetuity. (d) If the interest offered is 5.2 % per annum, compounded monthly, how much will Alicia receive per month, correct to the nearest dollar ? (1 mark) 0.43% x 625000 = 2687.50 = $ 2688 nearest dollar (0.052/12)x625000 = 2708.33 = $2708 nearest dollar (e) If Alicia wanted to receive at least $3000 per month from this perpetuity, how much would she need to invest, correct to the nearest thousand dollars? (2 marks) (3000*100)/(5.2/12) = 692307.7 = $692000 nearest thousand 697674.4186 = $ 698000 nearest thousand Page 5 of 10 PROBLEM THREE Jayde is a potter and has purchased a pottery kiln for $6500. The kiln can be depreciated using the reducing balance method at the rate of 17.5% per year. (a) Using Vn to represent the value of the kiln after n years, write a recurrence relation that models this depreciation situation. (2 marks) V0=6500 , Vn+1 = 0.825xVn (b) Use your calculator to determine recursively the value of the kiln, each year, for the first five years, and fill in the values in the table below. (2 marks) End of year Calculation details 0 Value of kiln 6500.00 1 .825x6500 5362.50 2 .825x5362.50 4424.06 3 .825x4424.06 3649.85 4 .825x3649.85 3011.13 5 .825x3011.13 2484.18 (c) If Jayde will trade in her kiln for a new one when this one has a value less than $2500, explain why this will occur after five years. After 5 years the kiln is worth $ 2484.18 which is below $2500 (1 mark) Jayde has been advised to look at using the prime cost or flat rate depreciation method using a value of 15% of purchase price. (d) By what amount, correct to the nearest dollar, will the value of the kiln be depreciated each year? (1 mark) 15% X 6500 =$975 (e) Using Vn to represent the value of the kiln after n years, write a recurrence relation that models this depreciation situation. (2 marks) V0=6500, Vn+1 = Vn - 975 V0=6500, Vn = 6500 - 975n (f) How many years will it take for the value of the kiln to depreciate to $2600? (1 mark) 4 years One of Jayde's pottery colleagues depreciates her kiln using the unit cost method at the rate of $5.70 per kiln load, where the average number of loads per year is 175. (g) Using Vn to represent the value of the kiln after n loads, write a recurrence relation that models this depreciation situation for Jayde. (2 marks) V0=6500, Vn+1 = Vn - 5.70 Page 6 of 10 V0=6500, Vn = 6500 - 5.70n (h) How long will it take in years, rounded to the nearest whole number, for the value of Jayde's kiln to depreciate to $2500 if it is depreciated using this method? (2 marks) 702 loads/175 loads per year =4.011 years round up to 5 years Mel is also accepting 4 years PROBLEM FOUR Sam has recently bought a home. He borrowed $350 000 from a bank, which is charging him 6.78 % per annum, compounding monthly, to be paid over twenty years. (a) What is the effective interest rate, correct to three significant figures, that Sam will be paying on his loan at 6.78 % per annum compounded monthly? (1 mark) r effective = 6.99% (b) Use the Finance Solver function/app on your calculator to determine Sam's monthly payment, correct to the nearest cent. Write the values used in the table below. (1 mark) N I(%) PV Pmt FV PpY, CpY 240 6.78 350000 -2667.5202 0 12 $2667.52 payment (c) What is the monthly interest rate that Sam will be charged, written as a decimal correct to three decimal places? (1 mark) 6.78/12 = 0.565% per month Sam's monthly payment is set at $2670.00 per month by his bank. (d) Using Bn to represent the balance of the loan after n months, write down a recurrence relation to model this loan situation. (2 marks) B0=350000, Bn+1 = 1.00565xBn - 2670 Part of the amortisation table for Sam's loan is shown below. (e) Payment number Payment made Interest paid 0 0.00 0.00 1 2670.00 2 2670.00 1973.59 3 2670.00 1969.65 4 2670.00 1965.70 5 2670.00 1961.72 6 2670.00 1957.71 ... ... ... Calculate, correct to the nearest cent: Principal reduction 0.00 692.50 700.35 704.30 708.28 712.29 ... (i) the interest that will be paid with the first payment 2670 - 692.50 = $1977.50 Page 7 of 10 Balance of loan 350 000.00 349 307.50 348 611.09 347 206.44 346 498.16 345 785.87 ... (1 mark) (ii) the amount that is reduced from the principal with the second payment 2670 - 1973.59 = $696.41 (1 mark) (iii) the balance of the loan after the third payment is made. 346611.09 - 700.35 = $347910.74 (1 mark) (f) (1 mark) Calculate the total amount of money paid by Sam for his first six payments. 6 x 2670 = $16020 in payments (g) How much has Sam actually paid off his loan after the sixth payment? (1 mark) 350000 - 345785.87 = $4214.13 off loan (h) Calculate the percentage, correct to the nearest whole number, of Sam's first six payments that have been used to pay the interest charges. 16020 - 4214.13= $11805.87 (i) (1 mark) 11805.87/16020 x 100% =73.6945% = 74% The loan is to be repaid with monthly payments of $2670.00 and a final payment that is to be adjusted so that the loan will be fully repaid after exactly 20 years of monthly payments. Calculate the amount of the final payment, correct to the nearest cent. (3 marks) Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY 240 6.78 350000 -2670.00 ? =1257.826 12 Positive FV means it's in Sam's favour, therefore final payment is reduced. Final Payment =2670.00 - 1257.83 = $1412.17 PROBLEM FIVE Trent purchased a new small car to use for his business at a cost of $38 000. He is allowed to depreciate it in value at a rate of 35 cents for every kilometre driven. (a) Using Vn to represent the value of the car after travelling n kilometres, write a recurrence relation that models this depreciation situation. (1 mark) V0=38000, Vn+1 = Vn - 0.35 V0=38000, Vn = 38000 - 0.35n (b) What is the value of the car after it has travelled 22 500 kilometres? (1 mark) $30125 (c) If the scrap value of the car is $15 000, how many kilometres will it have travelled, correct to the nearest hundred kilometre, to reach its scrap value? (1 mark) n = 2300/0.35 , n = 65714.28571 = 65700km Page 8 of 10 PROBLEM SIX An oil well started producing 1000 barrels of oil per day. The rate at which oil production reduces each day is called the decline rate and is used to predict the productive life of the oil well. (a) Output for the first three days was recorded as 1000, 980, 960.4. Calculate the decline rate for this oil well. (1 mark) 980/1000 = 960.6/980 = 0.98 = 2% decline . (b) State the recurrence relation for the oil production Pn on the nth day of production. (1 mark) P0 =1020.41, Pn+1 = 0.98xPn Pn = 0.98 n x1020.41 P1 =1000, Pn+1 = 0.98xPn Pn = 0.98 n-1 x1000 (c) What is the expected production for the well on the tenth day of operation? (1 mark) 833.74776 = 833.7 barrels (d) How many days of production will be possible before the daily output falls below 700 barrels? (1 mark) 19 days PROBLEM SEVEN Kate and Ben are managing a farm that is being used for beef cattle. At the beginning of their first year on the property they have 400 cattle. The rate of natural increase in the number of cattle is expected to be 8% per year and it is planned to sell 100 cattle at the end of each year. This situation can be given by the recursive equation T(n+1) = a T(n) + b where T(1) = c (is this the beginning or end yr1) where a, b and c are positive constants. (a) State the values of a, b and c. (1 mark) a=1.08, b= -100, c=400 (b) Use the recursive equation to determine the number of cattle in the herd at the beginning of the second year. (1 mark) 1.08x400 - 100 = 332 (c) Explain briefly why the plan to sell 100 head of cattle each year could not be kept going indefinitely. (1 mark) They are selling more cattle than they are replacing each year, so the would eventually have no cattle left. (d) How many cattle should be sold at the end of each year to keep the cattle population stable; that is, so that there are 400 cattle at the beginning of each year? (1 mark) D = 0.08x400 = 32 (e) Due to drought and disease, the rate of natural increase in the number of cattle drops to 6%. How many cattle must now be sold at the end of each year, so that there are 400 cattle at the beginning of each year? (1 mark) D = 0.06x400 = 24 Page 9 of 10 Page 10 of 10 Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC MULTIPLE CHOICE 1. C 11. A 21. E 2. A 12. A 22. E 3. 13. 23. D C B 4. D 14. A 24. C 5. E 15. D 25. A 6. B 16. C 26. D 7. E 17. E 27. D 8. B 18. E 9. D 19. B 10. E 20. E SESSION ONE PROBLEM ONE Kylie has invested $1200 in a special savings account that pays simple interest, with the interest being calculated quarterly. Using Vn to represent the value of the investment after n quarters, the recurrence relation that models this investment situation is : V0 = $1200 (a) Vn+1 = Vn + 9 Write down, in full, a calculation process, based on the recurrence relation, that shows that the interest rate applying to this investment is 3.0 % per annum. The value 9 represents the amount of interest paid for one quarter, so the interest paid for a full year will be 9 4 = $36 Interest rate = = 3.0 % 36 100 1200 (b) If the total value of Kylie's investment is $1326.00, for how long has the investment been running ? Using Vn = V0 + nD, where Vn = $1326.00, V0 = $1200, D = 9 1326 = 1200 + 9n 9n = 126 n = 14 This represents 14 quarters, or 3.5 years. PROBLEM TWO Lynne has invested $1500 in an account paying interest at 4.2 % per annum, compounded monthly. When asked, by her Mathematics teacher, to write a recurrence relation to model this investment, using Vn to represent the value of the investment after n months, she wrote the following: (i) V0 = $1500 Vn+1 = 0.042Vn When her teacher told her that this recurrence relation was incorrect, she wrote the following: (ii) V0 = $1500 Vn+1 = 1.042Vn When told that this recurrence relation was also incorrect, she eventually wrote the following, correct, recurrence relation : (iii) (a) V0 = $1500 Vn+1 = 1.0035Vn Explain why recurrence relation (i) is incorrect. May 2016 Page 1 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC The multiplying factor is incorrect, based on the annual interest rate May 2016 Page 2 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS (b) WSC Explain why recurrence relation (ii) is incorrect. The multiplying factor is incorrect, based on the annual interest rate (c) Use your calculator to determine recursively the value of the investment, correct to the nearest cent, each month, for the first six months, and complete the table below. Month number Calculation details Value of investment 0 (d) 1500.00 1 1500 1.0035 1505.25 2 1505.25 1.0035 (1510.5183...) 1510.52 3 1510.5183... 1.0035 (1515.8051...) 1515.81 4 1515.8051... 1.0035 (1521.1105...) 1521.11 5 1521.1105... 1.0035 (1526.4343...) 1526.43 6 1526.4343... 1.0035 (1531.7769...) 1531.78 Determine the amount of interest that Lynne has earned in the first six months, correct to the nearest cent. Interest = Amount - principal = $1531.78 - $1500.00 = $31.78 (e) If Lynne does not touch this investment for 18 months, how much will it be worth after this time, correct to the nearest cent ? Using Vn = Rn V0, where R = 1.0035, n = 18, V0 = 1500 Vn = 1.003518 1500 = $1597.3645... $1595.36 (f) If Lynne could earn 4.8 % per annum, compounded monthly, on her $1500 investment, how much more interest, correct to the nearest cent, could she have earned over the eighteen months compared to earning interest at 4.2 % per annum, compounding monthly ? Monthly Interest rate : = Recurrence relation : 4.8% 12 = 0.4 % = 0.004 % V0 = $1500 Vn+1 = 1.004Vn Using Vn = Rn V0, where R = 1.004, n = 18, V0 = 1500 Vn = 1.00418 1500 = $1611.7515... $1611.75 Difference = $1611.75 - $1595.36 = $16.39 May 2016 Page 3 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC SESSION TWO An astute borrower is keen to minimise the cost of his home loan. He is about to borrow $300 000 at 6.575 % p.a. interest, compounded monthly, and with monthly payments for twenty years. His accountant has advised him that by paying half his normal monthly repayment every fortnight, the term and total cost of his loan will be reduced. (a) Calculate the monthly payment required to amortise the loan, correct to the nearest dollar. Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY 240 6.575 300 000 ? 0 12 Solve for Pmt : 2249.9854... $2250 (b) Calculate the total paid over the term of the monthly payment loan. N I(%) PV Pmt FV PpY, CpY 240 6.575 300 000 -2250 ? 12 Calculate the residual payment Solve for FV : 7.18656... $7.19 Total paid = 240 2250 - 7.19 = $539 992.81 (c) Calculate the fortnightly payment required by halving the monthly payment. Fortnightly payment = $2250 2 = $1125.00 (d) Calculate the number of fortnightly payments needed to repay the loan, rounded up to the next whole number. Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY ? 6.575 300 000 - 1125.00 0 26,12 Solve for N : 443.6729... 444 (e) Calculate the total paid over the term of the fortnightly payment loan. N I(%) PV Pmt FV PpY, CpY 444 6.575 300 000 - 1125.00 ? 26,12 Calculate the residual payment Solve for FV : 367.65 May 2016 Page 4 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC Total paid = 444 1125 - 367.65 = $499 132.35 (f) Calculate the number of years and months needed to pay the loan with fortnightly payments. Number of years = 444 26 = 17.076... years = 17 years 1 month (g) Calculate the total amount of money saved over the term of the loan by the borrower paying fortnightly. Savings = $539 992.81 - $499 132.35 = $40 860.46 (h) Calculate the number of years and months of payments saved by paying fortnightly. Time saved = 20 years - 17 years 1 months = 2 years 11 months (i) the total amount paid each year by the monthly payment. Annual payment total = 12 $2250 = $27 000 (j) the total amount paid each year by the fortnightly payment. Annual payment total = 26 $1125 = $29 250 (k) Explain why this type of fortnightly loan is repaid quicker (and cheaper) than the monthly loan The loan was able to be repaid more quickly by being repaid fortnightly - nearly 3 years of payments being saved. The main reason for the quicker repayment was that an amount equal to a whole month's repayment was paid extra each year, because there are 26 fortnights in a year (compared to 24 half months). The secondary reason is that by paying fortnightly but compounding monthly you save a small amount of interest as the balance of the loan is reducing during the month, thereby reducing the amount of interest charged. May 2016 Page 5 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC SESSION THREE PROBLEM ONE (a) Stephen has borrowed $50 000 to invest in shares. He has taken out an interest-only loan with interest charged at 8.65 % per annum, compounded monthly. He will make monthly interest-only payments and pay back the full $50 000 after three years. Calculate the amount of the monthly payment, correct to the nearest cent. Interest (payment) = (b) 8.65 50000 100 12 = 360.41666... $360.42 If he makes $65 000 from the sale of his shares after three years, calculate how much profit or loss he makes overall on the deal. Total expenses = 36 $360.42 + $50 000 = $62 975.12 Profit = Income - expenses = $65 000 - $62 975.12 = $2024.88 (c) Stephen now wants to borrow money to help finance a building project. If his bank can offer an interest-only loan at 8.5 % per annum interest, how much could he apply for (to the nearest five thousand dollars), if he can afford a maximum monthly payment of $1500? principal = interest amount 100 12 rate = 1500 100 12 8.5 $211 764.705... He can apply for $210 000 to keep the monthly repayment UNDER $1500 !! (d) What is the effective rate of interest, correct to two decimal places, that Stephen will be paying on his interest-only loan at 8.5 % per annum, compounded monthly? reffective = May 2016 8.5 1 100 12 12 1 100% = 8.83909... 8.84 % Page 6 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC PROBLEM TWO (a) Roger wants to establish a scholarship fund to support students at his old secondary college. He plans to invest $125 000 in special bonds that pay 4.5 % per annum interest. If the interest is calculated and paid annually as the scholarship amount, how much money will be available each year to be awarded ? Interest (payment) = (b) 4.5 125000 100 = $5625.00 Roger actually wants to have at least $7500 available each year. How much money will he have to invest, to the nearest thousand dollars, at 4.5 % per annum to earn the required sum each year ? principal = interest amount 100 rate = 7500 100 4.5 $166 666.666... He will have to invest $167 000 to meet the interest requirement. (Note : if he invests $166 000, he will earn less than $7500 per year - $7470 !) May 2016 Page 7 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC PROBLEM THREE Nick is self-employed as a sub-contract carpenter and has decided to set up a superannuation fund. He has started the fund with $10 000 he has saved and invested it in an account earning 5.70 % per annum, compounded monthly. He has decided to add $500 each month to the fund. (a) Use your calculator to determine recursively the value of the superannuation fund, correct to the nearest cent each month, for the first six months, and complete the table below. Month number Calculation details Value of fund 0 (b) 10 000.00 1 10 000 1.00475 + 500 10 547.50 2 10 547.50 1.00475 + 500 (11 097.6006...) 11 097.60 3 11 097.6006... 1.00475 + 500 (11 650.3142...) 11 650.31 4 11 350.3142... 1.00475 + 500 (12 205.6532...) 12 205.65 5 12 205.6532... 1.00475 + 500 (12 763.6300...) 12 673.63 6 12 763.6300... 1.00475 + 500 (13 324.2573...) 13 324.26 If he can make monthly payments of $500 for the next thirty years, how much will be in his fund when he retires ? Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY 360 5.70 -10 000 - 500 ? 12 Solve for FV : 529 447.608... (c) He will have $529 447.61. Nick believes that he can make a success of his business and he will be able to add more money to his superannuation fund over time. He plans to pay $500 per month for the first ten years, $600 per month for the next ten years and $700 per month for the last ten years. If he continues to receive 5.70 % per annum, how much money will be in his superannuation fund after these thirty years? Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY 120 5.70 -10 000 - 500 ? 12 Solve for FV : 98 278.026... May 2016 He will have $98 278.03 after ten years. Page 8 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY 120 5.70 - 98 278.03 - 600 ? 12 Solve for FV : 270 290.488... He will have $270 290.49 after twenty years. Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY 120 5.70 - 270 290.49 - 700 ? 12 Solve for FV : 590 168.108... May 2016 He will have $590 168.11 after thirty years. Page 9 of 10 Dalmau/McMahon/Tschiderer Core 2: Recursion & Financial modelling - Practise SAC SOLUTIONS WSC PROBLEM FOUR Jennifer now works from home as a freelance graphic designer after a number of years working in large organisations. She currently has $125 000 in her superannuation account, but her accountant advises her that she will need to have at least $650 000 in the account when she plans to retire in twenty years. If her superannuation account is earning 5.4 % interest per annum, compounded monthly, how much will Jennifer have to add each month, rounded to the nearest dollar, to at least meet her intended goal in twenty years ? Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY 300 5.4 -125 000 ? 650 000 12 Solve for Pmt : - 632.241... She will have to contribute $633 per month (which will give her $650 330) May 2016 Page 10 of 10 Dalmau/McMahon/Tschiderer Recursion and financial modelling Part 1: Short Answer Response. 1.Ian has perpetuity that earns interest of 3.8% p.a compounding monthly, what is the interest earnt in the 6th month. With an initial investment of $190750, to the nearest dollar? 1.a. After 5 years how much money does Ian have in his perpetuity? 2) Sally has an investment of $2000, paying interest at the rate of 9% per annum, compounding monthly, write a recurrence relation that can be used to model sally interest. 3) In a geometric sequence, what is the common ratio that will result in a regular decrease of 9%? Answer to two decimal places. 4) What is the rate for the nth of the recursion equation given by T 0=10, Tn+1= Tn + 8? 5) A sequence is defined by the recursion equation T 0= x, tn + 1= 4tn - I. If t3 is 171, what is the value of x? 6) Brian invested $50000 into an account that earn compound interest at the rate of 4.6% per annum, compounding fortnightly. After one year, the value of the investment has grown to $58997 what is the amount Brain is adding to his investment every fortnightly. 7) The recurrence relation V0= 10000, Vn + 1 = 16012vn-2000 models a reducing - balance loan with payments of $2000 each quarter, where Vn is the value of the loan after n questers. Write down the annual interest rate for this loan? Part 2: Extended Answer Response. (PLZ SHOW ALL WORKING OUT) 1. Louise inherited an oil well, the day before she finally talks ownership of it, the equipment was valued at $100000, however has been reducing in value due to wear and tear ever since. The rate at which the equipment value decays is called the depreciation rate and is used to predict the production life of the oil well equipment. a) Yearly estimated values were recorded as $1000000, $980000, $ 960400. Calculate the depreciation rate for this oil equipment as a percentage to 1 decimal place. b) State the recurrence relation for the oil production pn on the nth day of production. c) What is the expected value for the equipment on the tenth year of opening after she ownership, nearest thousand and dollars. d) How many years of production will be possible before the equipment value falls below $700000 2. Phillip is a chef. He runs a cooking school from his home kitchen. A 10-week course will cost $1200 if paid up front Boon Soo enrolled in the course, but has negotiated to borrow the course fee from Phillip and pay it back with $2- interest each week. Len Vn be the value of Boon Soo loan after n weeks. a) Write down a recurrence relation that made models the value of this simple- interest loan. After n weeks. b) Use the recurrence relation to find the amount that Boon Soo will owe Phillip after 4 weeks. The rule un= 1200 + 20 n can be used to calculate the value of Boon Soo's loan after n weeks. C) What is the value of Boon Soo's loan after 6 weeks? d) Use the rule to calculate the number of weeks it teaches, the value of Boon's Soo's loan to grow to $1360. Part 3: Real Life questions (PLZ SHOW ALL WORKING OUT) Bill is doing some research, on loans available for a 30-year period, compounding monthly. Loans with a lower rate for the first year; fixed rate loans that revert to variable, Home loan packages with lower rates. The standard variable rate is 4.64% p.a Bill is buying a house for $295000. Package variable rate loan: no establishment fee but has an annual fee of $395 and has 0.8% off the standards variable rate while the loan is above $250000 and 0.6% discount while the loan is above $150000. Flexi First Option Home Loan: A discount of 0.65% for 3 years, then reverts to the standard variable rate and has no establishment fee. Package Flexi First Option Home Loan: A further discount of 0.65% over the Flexi First Option Home Loans for 3 years, then reverts to the standard variable rate and has no establishment fee but has an annual fee of $395. Fixed rate option: $600 application fee, reverts to variable rate when completed. Fixed Period Interest Rate 1 Year 4.59% 2 Year 4.19% 3 Year 4.29% 4 Year 4.79% 5 Year 4.59% Fixed Rate loans: same as fixed but with the rate discounted by 0.2% as long as the balance completed is $150000, however has no application fee but does have an annual fee of $395. 1. Calculate the total cost Flexi First Option Home Loan for monthly and fortnightly payment given the bank cannot accept payments lower than half the normal monthly payment and that Bill always pay the minimum payment required. Show any details of Financial Solver Application and include allowances for all fee and residual payment (if any) Flexi First Option Home Loan: Monthly: N I (%) PV Pmt FV PpY CpY PV Pmt FV PpY CpY PV Pmt FV PpY CpY PV Pmt FV PpY CpY PV Pmt FV PpY CpY Pmt FV PpY CpY Pmt FV PpY CpY FV after 3 years of pmt's N I (%) New Monthly Pmt's N I (%) Residue (FV) N I (%) FORTNIGHTLY: Monthly Pmt's N I (%) Number of fortnightly Pmt's N I (%) PV FV after 3 years of Pmt's N I (%) PV New Monthly Pmt's N I (%) PV Pmt FV PpY CpY PV Pmt FV PpY CpY PV Pmt FV PpY CpY Number of fortnightly pmt's N I (%) Residue (FV) N I (%) Total Cost: Recursion and financial modelling Part 1: Short Answer Response. 1.Ian has perpetuity that earns interest of 3.8% p.a compounding monthly, what is the interest earnt in the 6th month. With an initial investment of $190750, to the nearest dollar? 1.a. After 5 years how much money does Ian have in his perpetuity? 2) Sally has an investment of $2000, paying interest at the rate of 9% per annum, compounding monthly, write a recurrence relation that can be used to model sally interest. 3) In a geometric sequence, what is the common ratio that will result in a regular decrease of 9%? Answer to two decimal places. 4) What is the rate for the nth of the recursion equation given by T 0=10, Tn+1= Tn + 8? 5) A sequence is defined by the recursion equation T 0= x, tn + 1= 4tn - I. If t3 is 171, what is the value of x? 6) Brian invested $50000 into an account that earn compound interest at the rate of 4.6% per annum, compounding fortnightly. After one year, the value of the investment has grown to $58997 what is the amount Brain is adding to his investment every fortnightly. 7) The recurrence relation V0= 10000, Vn + 1 = 16012vn-2000 models a reducing - balance loan with payments of $2000 each quarter, where Vn is the value of the loan after n questers. Write down the annual interest rate for this loan? Part 2: Extended Answer Response. (PLZ SHOW ALL WORKING OUT) 1. Louise inherited an oil well, the day before she finally talks ownership of it, the equipment was valued at $100000, however has been reducing in value due to wear and tear ever since. The rate at which the equipment value decays is called the depreciation rate and is used to predict the production life of the oil well equipment. a) Yearly estimated values were recorded as $1000000, $980000, $ 960400. Calculate the depreciation rate for this oil equipment as a percentage to 1 decimal place. b) State the recurrence relation for the oil production pn on the nth day of production. c) What is the expected value for the equipment on the tenth year of opening after she ownership, nearest thousand and dollars. d) How many years of production will be possible before the equipment value falls below $700000 2. Phillip is a chef. He runs a cooking school from his home kitchen. A 10-week course will cost $1200 if paid up front Boon Soo enrolled in the course, but has negotiated to borrow the course fee from Phillip and pay it back with $2- interest each week. Len Vn be the value of Boon Soo loan after n weeks. a) Write down a recurrence relation that made models the value of this simple- interest loan. After n weeks. b) Use the recurrence relation to find the amount that Boon Soo will owe Phillip after 4 weeks. The rule un= 1200 + 20 n can be used to calculate the value of Boon Soo's loan after n weeks. C) What is the value of Boon Soo's loan after 6 weeks? d) Use the rule to calculate the number of weeks it teaches, the value of Boon's Soo's loan to grow to $1360. Part 3: Real Life questions (PLZ SHOW ALL WORKING OUT) Bill is doing some research, on loans available for a 30-year period, compounding monthly. Loans with a lower rate for the first year; fixed rate loans that revert to variable, Home loan packages with lower rates. The standard variable rate is 4.64% p.a Bill is buying a house for $295000. Package variable rate loan: no establishment fee but has an annual fee of $395 and has 0.8% off the standards variable rate while the loan is above $250000 and 0.6% discount while the loan is above $150000. Flexi First Option Home Loan: A discount of 0.65% for 3 years, then reverts to the standard variable rate and has no establishment fee. Package Flexi First Option Home Loan: A further discount of 0.65% over the Flexi First Option Home Loans for 3 years, then reverts to the standard variable rate and has no establishment fee but has an annual fee of $395. Fixed rate option: $600 application fee, reverts to variable rate when completed. Fixed Period Interest Rate 1 Year 4.59% 2 Year 4.19% 3 Year 4.29% 4 Year 4.79% 5 Year 4.59% Fixed Rate loans: same as fixed but with the rate discounted by 0.2% as long as the balance completed is $150000, however has no application fee but does have an annual fee of $395. 1. Calculate the total cost Flexi First Option Home Loan for monthly and fortnightly payment given the bank cannot accept payments lower than half the normal monthly payment and that Bill always pay the minimum payment required. Show any details of Financial Solver Application and include allowances for all fee and residual payment (if any) Flexi First Option Home Loan: Monthly: N I (%) PV Pmt FV PpY CpY PV Pmt FV PpY CpY PV Pmt FV PpY CpY PV Pmt FV PpY CpY PV Pmt FV PpY CpY Pmt FV PpY CpY Pmt FV PpY CpY FV after 3 years of pmt's N I (%) New Monthly Pmt's N I (%) Residue (FV) N I (%) FORTNIGHTLY: Monthly Pmt's N I (%) Number of fortnightly Pmt's N I (%) PV FV after 3 years of Pmt's N I (%) PV New Monthly Pmt's N I (%) PV Pmt FV PpY CpY PV Pmt FV PpY CpY PV Pmt FV PpY CpY Number of fortnightly pmt's N I (%) Residue (FV) N I (%) Total Cost: Still Part 3: This continues from the previous work you did on Part 3. Encourage to show working out 1. Calculate the total cost for the fixed rate loans for monthly payments, given the bank cannot accept payments lower than half the normal monthly payment and that Bill will always pay the minimum payment required. Show any details of financial solver application and include allowances for all fees and residual payments (if any) Fixed rate option (4 year) Monthly N I (%) PV Pmt FV PpY CpY N I (%) PV Pmt FV PpY CpY N I (%) PV Pmt FV PpY CpY N I (%) PV Pmt FV PpY CpY Total Cost: 2. Considered loan options calculated for Bill, which is the best home loan option from Bill's research? Include your respond. 3. After 5 years, Bill receieves a winddan of $30000 in the form of an inheritance. Bill plans on paying a lump sum of 30000 off the package flexi first option home loan with monthly repayments, in terms of reducing the time the loan takes to pay off. Assume the monthly repayments were kept the same once the lump sum had reduces the loan balance. What is the final residual payment, total number of repayments and the new total cost of the loan? Monthly: N I (%) PV Pmt FV PpY CpY N I (%) PV Pmt FV PpY CpY N I (%) PV Pmt FV PpY CpY N I (%) PV Pmt FV PpY CpY N I (%) PV Pmt FV PpY CpY N I (%) PV Pmt FV PpY CpY - Total time in years and months. Residual payment: Total cost: Part 2: Extended response. 1. Jayde has purchased a car for $8500. The car can be depreciated using the reducing balance method at the rate of 14.5% per year. Use your calculator to determine recursively the value of the car, each year, for the first five years, and fill in the values in the table below. End of year 0 1 2 3 4 5 Calculation details Value of car 8500.00 Has been advised to look at using the prime cost or flat rate depreciation method using a value of 15% purchase price. a) By what amount, correct to the nearest dollar, will the value of the car be depreciated each year? b) Using Vn to represent the value of the car after n years, write a recurrence relation the models this depreciation situation. c) How many years will it take for the value of the car to depreciate to $4500? 2. Sam has recently bought a home. He borrowed $450000 from a bank, which is charging him 6. 23% per annum, compounding monthly, to be paid over twenty years. a) What is the effective interest rate, correct to three significant figures, that Sam will be paying on his loan at 6.23% per annum compounded monthly? b) Use the Finance Solver function/app on your calculator to determine Sam's monthly payment, correct to the nearest cent. Write the values used in the table below. N I(%) PV Pmt FV PpY, CpY c) What is the monthly interest rate that Sam will be charged, written as a decimal correct to three decimal places? Sam's monthly payment is set a $3285.00 per month by his bank. d) Using Bn, to represent the balance of the loan after n months, write down a recurrence relation to model this loan situation. Part of the amortisation table for Sam's loan is shown below. Payment No. PMT Interest Paid 0 1 2 3 4 5 6 0.00 3285.00 3285.00 3285.00 3285.00 3285.00 3285.00 0.00 2330.57 2325.62 2320.64 2315.63 2310.60 Principal Reduction 0.00 949.50 959.38 964.36 969.37 974.40 Balance 450000.00 449050.50 448096.07 446172.33 445202.96 444228.57 Calculate, correct to the nearest cent: e) The interest that will paid with the first payment (ii) The amount that is reduce from the principal with the second payment (iii) The balance of the loan after the third payment is made. f) Calculate the total amount of money paid by Sam for his first sixth payment? g) How much has Sam actually paid off his loan after the sixth payment? h) Calculate the percentage, correct to the nearest whole number, of Sam's first six payments that have been used to pay off the principal. i) The loan is to be repaid with monthly payments of $3285.00 and a final payment that is to be adjusted so that the loan will be fully repaid after exactly 20 years of monthly payments. Calculate the amount of the final payment, correct to the nearest cent. Show details of any Finance Solver calculations in the table provided below. N I(%) PV Pmt FV PpY, CpY COMPLETED! GREAT JOB AND THANK YOU A $5 Tip will be provided for your effort and work