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11.Due east: Compound Involvement- Annuities (Exercises)

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11.i: Fundamentals of Annuities

Mechanics

For questions 1–4, use the information provided to determine whether an annuity exists.

1. A debt of four payments of $500 due in 6 months, 12 months, eighteen months, and 24 months.
2. A debt of iv quarterly payments in the amounts of $100, $200, $300, and $400.
3. Contributions to an RRSP of $200 every month for the first year followed by $200 every quarter for the second year.
4. Regular monthly deposits of $250 to an RRSP for v years, skipping ane payment in the 3rd year.

For questions 5–8, determine the annuity type.

Compounding Frequency	Payment Frequency	Payment Timing
5. Quarterly	Semi-annually	Get-go
6. Annually	Annually	End
7. Semi-annually	Semi-annually	Beginning
viii. Monthly	Quarterly	End

For questions 9–10, draw an annuity timeline and determine the annuity type.

9. A $2,000 loan at seven% compounded quarterly is taken out today. Four quarterly payments of $522.07 are required. The commencement payment will be 3 months afterward the first of the loan.
10. A new RRSP is set up with monthly contributions of $300 for five years earning nine% compounded semi-annually. The RRSP volition take $22,695.85 when complete. The outset payment is today.

Applications

For questions 11–15, draw an annuity timeline and determine the annuity type. Calculate the value of N.

11. Marie has decided to start saving for a down payment on her domicile. If she puts $i,000 every quarter for five years into a GIC earning 6% compounded monthly she will have $20,979.12. She will make her kickoff deposit three months from at present.
12. Laroquette Holdings needs a new $10 million warehouse. Starting today the company will put aside $139,239.72 every month for five years into an annuity earning 7% compounded semi-annually.
13. Brenda volition lease a $25,000 machine at 3.ix% compounded monthly with monthly payments of $473.15 starting immediately. After iii years she will still owe $10,000 on the vehicle.
14. Steve takes out a 2-yr gym membership worth $500. The first of his monthly $22.41 payments is due at signing and includes interest at 8% compounded annually.
15. Each year, Buhler Industries saves up $1 million to distribute in Christmas bonuses to its employees. To exercise so, at the end of every month the company invests $81,253.45 into an account earning 5.v% compounded monthly.

For questions 16–xx, assign the data in the timeline to the correct variables and determine the annuity type. Summate the value of N.

sixteen.

17.

xviii.

xix.

20.

11.ii: Future Value Of Annuities

Mechanics

For questions ane–four, calculate the future value.

Present Value	Interest Charge per unit	Payments	Timing of Payment	Years
1. $0	7% quarterly	$two,000 quarterly	Get-go	x
ii. $0	9% monthly	$375 monthly	Finish	20
3. $15,000	5.6% quarterly	$iii,000 annually	Finish	thirty
iv. $38,000	8% semi-annually	$one,500 monthly	Get-go	8

For questions v–8, calculate the future value.

Present Value	Interest Rate	Payments	Timing of Payment
five. $0	6% annually for four years; then 7% semiannually for six years	$one,000 quarterly	End
six. $0	12% quarterly for 6-and-a-half years; then 11% semi-annually for iii-and-a-half years	$100 monthly	Beginning
7. $27,150	xi% quarterly	$750 quarterly for five years; then $1,000 quarterly for 10 years	End
viii. $50,025	8% annually for three years; then 10% annually for seven years	$five,000 annually for iii years; and then $4,000 annually for vii years	Beginning

Applications

9. Nikola is currently 47 years one-time and planning to retire at age 60. She has already saved $220,000 in her RRSP. If she continues to contribute $200 at the start of every month, how much money will be in her RRSP at retirement if it tin can earn 8.1% compounded monthly? No deposit is made the solar day she turns 60.
10. You are a financial adviser. Your customer is thinking of investing $600 at the end of every half dozen months for the next six years with the invested funds earning 6.4% compounded semi-annually. Your customer wants to know how much coin she will accept after six years. What do you tell your client?
11. The Saskatchewan Roughriders started a rainy mean solar day savings fund iii-and-a-one-half years ago to help pay for stadium improvements. At the offset of every quarter the team has deposited $20,000 into the fund, which has been earning iv.85% compounded semi-annually. How much money is in the fund today?
12. McDonald's major distribution partner, The Martin-Brower Company, needs at least $1 million to build a new warehouse in Medicine Hat two years from today. To date, it has invested $500,000. If it continues to invest $fifty,000 at the stop of every quarter into a fund earning half-dozen% quarterly, volition it take enough money to build the warehouse two years from now? Bear witness calculations to support your answer.
13. The homo resources department helps employees save by taking preauthorized RRSP deductions from employee paycheques and putting them into an investment. For the first five years, Margaret has had $50 deducted at the outset of every biweekly pay period. So for the side by side five years, she increased the deduction to $75. The company has been able to average eight.85% compounded monthly for the first vii years, and then 7.35% compounded monthly for the last three years. What amount has Margaret accumulated in her RRSP after 10 years? Assume at that place are 26 biweekly periods in a year.
14. Joshua is opening up a Architect GIC that allows him to make regular contributions to his GIC throughout the term. He will initially deposit $x,000, then at the end of every month for the next five years he volition make $100 contributions to his GIC. The annually compounded interest rates on the GIC in each year are 0.75%, 1.5%, two.five%, four.5%, and 7.25%. What is the maturity value of his GIC?
15. How much more than money would an individual who makes $300 contributions to her RRSP at the beginning of the month have compared to another individual who makes $300 contributions to his RRSP at the end of the month? Assume a term of 30 years and that both RRSPs earn 9% annually.

Challenge, Critical Thinking, & Other Applications

16. When Shayla turned v years old, her mother opened up a Registered Education Savings Programme (RESP) and started making $600 end-of-quarter contributions. The RESP earned seven.46% semi-annually. At the terminate of each year, Human Resource and Skills Development Canada (HRSDC) fabricated an additional deposit under the Canada Educational activity Savings Grant (CESG) of 20% of her annual contributions into her RESP. Calculate the total maturity value available for Shayla's education when she turns 18.
17. Assume a 10-twelvemonth ordinary annuity earning 10% compounded annually.
    1. If $5,000 is deposited annually, what is the maturity value?
    2. What is the maturity value if the deposits are doubled to $10,000? Compared to (a), what is the human relationship betwixt the size of the deposit and the maturity value, all other conditions existence held equal?
    3. What is the maturity value if the $5,000 deposits are made semi-annually? Compared to (b), what is the relationship betwixt the frequency of payments and the maturity value, all other conditions being held equal?
18. Carlyle plans to make month-finish contributions of $400 to his RRSP from historic period twenty to age twoscore. From historic period 40 to age 65, he plans to make no farther contributions to his RRSP. The RRSP can earn ix% compounded annually from historic period 20 to age 60, and and then v% compounded annually from age 60 to age 65. Nether this programme, what is the maturity value of his RRSP when he turns 65?
19. To demonstrate the power of compound involvement on an annuity, examine the principal and interest components of the maturity value in your RRSP later on a certain time menses. Suppose $200 is invested at the end of every month into an RRSP earning 8% compounded quarterly.
    1. Decide the maturity value, principal portion, and involvement portion at ten, 20, 30, and 40 years. What do y'all observe?
    2. Change the interest rate to nine% quarterly and repeat (a). Comparing your answers to (a), what do you observe?
20. Compare the following maturity values on these annuities due earning 9% compounded semi-annually:
    1. Payments of $1,000 quarterly for xl years.
    2. Payments of $1,600 quarterly for 25 years.
    3. Payments of $4,000 quarterly for x years.

Note in all iii of these annuities that the same amount of principal is contributed. What can you learn about compound involvement from these calculations?

11.3: Present Value Of Annuities

Mechanics

For questions 1–three, calculate the corporeality of coin that must be invested today for an private to receive the future payments indicated and take the remaining balance at the end of the term.

Future Value	Interest Rate	Payments	Timing of Payment	Years
one. $0	7% quarterly	$2,000 quarterly	Beginning	10
ii. $250,000	5.vi% quarterly	$iii,000 annually	End	30
3. $380,000	eight% semi-annually	$one,500 monthly	Kickoff	8

For questions 4–six, calculate the amount of money that must be invested today for an individual to receive the time to come payments indicated and have the remaining balance at the end of the term.

Futurity Value	Interest Rate	Payments	Payment Timing
four. $0	7% semi-annually for half dozen years; then 6% annually for four years	$one,000 quarterly	End
5. $327,150	11% quarterly	$1,000 quarterly for ten years; then $750 quarterly for five years	End
6. $150,025	10% annually for seven years; and so 8% annually for 3 years	$4,000 annually for seven years; then $5,000 annually for iii years	Beginning

For questions 7–eight, summate the remainder owing and total interest paid over the time menstruum indicated (from the start) for the following ordinary loans.

Initial Loan Amount	Interest Rate	Payments	Balance Owing Afterwards
seven. $35,000	8% quarterly	$853.59 monthly	Two years, half-dozen months
8. $48,000	9% monthly	$865.23 monthly	Four years, eight months

For questions nine–10, calculate the proceeds of the auction for the following sales of ordinary loan contracts.

Time Left on Loan on Engagement of Sale	Payments	Last Payment Amount	New Negotiated Interest Charge per unit
9. Three-and-a-half years	$ane,655.74 semi-annually	$1,655.69	14.85% monthly
10. 4-and-a-quarter years	$1,126.96 monthly	$1,127.21	12.9% quarterly

Applications

11. When Sinbad retires, he expects his RRSP to pay him $2,000 at the end of every calendar month for 25 years. If his retirement annuity earns three.8% compounded quarterly, how much money does he need to have in his RRSP when he retires?
12. Sandy's parents would like to have an annuity pay her $500 at the beginning of every calendar month from September 1, 2012, to April 1, 2017, to assistance with her university tuition and living expenses. On May 1, 2017, they would like to give her a graduation gift of $v,000. If the annuity can earn 6.xv% compounded monthly, how much money must be in the account on September i, 2012?
13. The Workers' Compensation Board has determined that an injury in the workplace was your company's responsibility. As a result, your company has been ordered to pay the employee $three,000 at the stop of every month for the side by side four years. Your human resource manager wants to set upwardly an annuity to fund this obligation. If the proposed annuity tin can earn 5.7% compounded monthly for the showtime two-and-a-half years and then 6% compounded quarterly for the remaining one-and-a-one-half years, how much money should your company set aside today to meet its responsibilities?
14. Working in the accounting section, Jaycee needs to accurately tape the debts of the company. Nine months ago, the company purchased new production equipment for $88,437.48 and financed it on a 12-month loan at viii.ii% compounded quarterly. The payments at the terminate of every month have been $seven,698.95. What amount should Jaycee record as the balance attributable today? How much interest has been paid to appointment?
15. Sleep Country Canada completed a auction of an entire mattress and box jump gear up to a client for $2,250 to be paid in 12 equal month-cease instalments with no interest. If it immediately sells this contract on the engagement of result to CitiFinancial at 12% compounded annually, what are the proceeds of the auction?
16. Three years and 2 months ago, Mr. Magoo purchased a brand new Volkswagen Highline Jetta in Toronto for $32,854.75. He paid $5,000 as a downwards payment and financed the rest at 0.9% compounded monthly for half-dozen years. His payments have been $397.56 at the end of every month.
    1. What is the balance still attributable on his vehicle today?
    2. If the dealership sells the loan contract today to a finance visitor at 9.ix% compounded monthly, what are the proceeds of the sale? The last payment is for $397.85.

Challenge, Disquisitional Thinking, & Other Applications

17. Lynne acquired a Body of water Ray Sundancer boat and put $4,000 down. For the past two years, her end-of-month payments accept been $1,049.01 including nine.32% compounded monthly. If she nonetheless owes $22,888.78 today, what was the purchase price of the gunkhole?
18. Gerald has been granted power of chaser and is now responsible for setting up his aging parents in a seniors' home. The rent volition be $2,490 at the showtime of every month for the first year, then increase by v% the following year and four% in the tertiary year. Gerald wants to take money from his parents' estate and ready an annuity to pay their monthly rent. If he can go an annuity that earns iii.75% semi-annually, how much money from his parents' estate needs to be invested today to see the rental payments over the next 3 years?
19. Compare the corporeality of money that needs to be invested today to provide the required payments from the investment fund annuities earning nine% compounded semi-annually:
    1. Payments of $i,000 quarterly for twoscore years.
    2. Payments of $1,600 quarterly for 25 years.
    3. Payments of $4,000 quarterly for 10 years. Note that in all three of these annuities the aforementioned total payout occurs. Explain your results and comment on your findings.
20. HSBC Finance Canada is going to purchase the following ordinary loan contracts from the same company on the same engagement. In all cases, HSBC demands an interest rate of 18.9% compounded monthly on its purchases. What are the total proceeds of the sales?
    1. $734.56 quarterly payments with four years remaining in the term, and the final payment is $734.64.
    2. $i,612.46 semi-almanac payments with six-and-a-half years remaining, and the concluding payment is $1,612.39.
    3. $549.98 monthly payments with five years and two months remaining, and the terminal payment is $550.28.

11.4: Annuity Payment Amounts

Mechanics

For questions 1–8, calculate the annuity payment amount.

Annuity Value	Single Payment Amount	Term of Annuity	Payment Frequency	Nominal Interest Rate	Compounding Frequency
i. \(FV_{DUE}\) = $25,000	$0	v years	Monthly	viii.25%	Monthly
2. \(PV_{ORD}\) = $500,000	$0	fifteen years	Quarterly	5.ix%	Semi-annually
iii. \(PV_{DUE}\) = $one,000,000	$0	25 years	Monthly	4.75%	Semi-annually
4. \(FV_{ORD}\) =$i,500,000	$0	35 years	Annually	9%	Annually
v. \(PV_{ORD}\) = $50,000	\(FV\) = $5,000	6½ years	Semi-annually	iii.65%	Quarterly
vi. \(FV_{ORD}\) =$five,000,000	\(PV\) = $450,000	iv years	Annually	4.35%	Monthly
7. \(FV_{DUE}\) = $1,000,000	\(PV\) = $5,000	40 years	Monthly	7.92%	Quarterly
8. \(PV_{DUE}\) =$l,000	\(FV\) =$10,000	5 years	Monthly	5.5%	Annually

Applications

9. To salve approximately $xxx,000 for a downward payment on a home iv years from today, what amount needs to be invested at the end of every month at 4.5% compounded monthly?
10. At historic period 60, Tiger has managed to salve $850,000 and decides to retire. He wants to receive equal payments at the beginning of each month for the next 25 years. The annuity can earn v.4% compounded quarterly.
    1. If he plans on depleting the annuity, how much are his monthly payments?
    2. If he wants to have $50,000 left over at the end of the annuity, how much are his monthly payments?
11. To purchase his new car, Scooby-Doo has obtained a half dozen-year loan for $40,000 at viii.8% compounded semi-annually.
    1. Determine the monthly payments required on the loan.
    2. Calculate the remainder owing and the full interest paid after 3 years of making payments toward the loan.
    3. Instead of (b), Scooby-Doo is because reducing the rest owing to $15,000 later 3 years and paying off the loan in full at that fourth dimension. What monthly payments are required?
12. Aureate's Gym wants to offer its clients a monthly payment option on its annual membership dues of $490. If the gym charges seven.75% compounded quarterly on its membership fee, what showtime-of-calendar month payments should it advertise?
13. A 20-year marketable bail can be purchased today for $13,402.90. It will make interest payments to the investor at the stop of every 6 months, along with a $ten,000 lump-sum payment to the investor at the cease of the term. If prevailing interest rates are vi.85% compounded semi-annually, how much is each interest payment?
14. Carling Industries needs to larn some real estate to expand its operations. In negotiations with the Province of Nova Scotia, it will be immune to purchase the $15 meg parcel of state today and commencement making payments at the stop of every six months for the adjacent 10 years. If interest volition be charged at seven.6% compounded semi-annually, what volition be the required payments? (Circular to the nearest dollar.)
15. Sinclair does not believe in debt and will simply pay cash for all purchases. He has already saved up $140,000 toward the buy of a new domicile with an estimated cost of $300,000. Suppose his investments earn 7.5% compounded monthly. How much does he need to contribute at the showtime of each quarter if he wants to purchase his home in five years?
16. A-Ane Courier Services needs to lease five vehicles for the adjacent iii years. Each vehicle retails for $23,750, and the interest rate on the lease is 5.85% compounded monthly. Under the lease terms, the company will make quarterly payments starting today such that the residue owing on each vehicle will be $10,300 at the end of the lease. Calculate the required lease payments.

Challenge, Disquisitional Thinking, & Other Applications

17. A sales representative tells a production managing director that if she purchases a new piece of machinery with a ii-year life expectancy for $twoscore,000 she will see a substantial reduction in operating costs. To purchase the machine, the product director will need to obtain a two-twelvemonth loan at 8% compounded quarterly. What is the to the lowest degree amount by which the monthly operating costs would need to be reduced for this purchase to brand economic sense? Presume that operating costs are reduced at the starting time of each month.
18. The Kowalskis' just child is eight years sometime. They desire to start saving into an RESP such that their son volition exist able to receive $5,000 at the end of every quarter for 4 years once he turns 18 and starts attending postsecondary schoolhouse. When the annuity is paying out, it is forecast to earn four% compounded monthly. While they make contributions at the end of every month to the RESP, information technology will earn viii% compounded semi-annually. Additionally, at the end of every yr of contributions the government places a $500 grant into the RESP. What is the monthly contribution payment by the Kowalskis?
19. Santana wants his retirement money to pay him $three,000 at the showtime of every month for 20 years. He expects the annuity to earn vi.xv% compounded monthly during this time. If his RRSP can earn 10.25% compounded annually and he contributes for the adjacent 30 years, how much coin does he need to invest into his RRSP at the end of every month? He has already saved $15,000 to engagement.
20. A lot of people fail to sympathize how interest rate changes affect their mortgages. Many call up that if the interest rate on their mortgage rises from v% to half-dozen% their payments will rise past 1%. Presume a $100,000 mortgage with stop-of-month payments for 25 years. Calculate the monthly mortgage payment at different semi-annually compounded involvement rates of 4%, 5%, half-dozen%, vii%, and 8%. What happens as the interest rate rises by 1% each fourth dimension?

11.v: Number Of Annuity Payments

Mechanics

For questions one–eight, calculate the number of annuity payments required and limited in a common date format.

Present Value	Time to come Value	Interest Rate	Annuity Payments	Payment Timing
i. $0	$100,000	vi.35% Quarterly	$iii,000 Quarterly	End
ii. $0	$500,000	four.viii% Monthly	$one,000 Monthly	Beginning
3. $0	$250,000	5.6% Semi-annually	$7,500 Annually	Cease
4. $0	$175,000	seven% Annually	$two,500 Semi-annually	Beginning
5. $300,000	$0	4.55% Semi-annually	$18,000 Semi-annually	End
6. $100,000	$0	6.5% Annually	$10,000 Annually	Start
seven. $50,000	$0	7.two% Quarterly	$500 Monthly	End
8. $i,000,000	$0	9% Monthly	$xl,000 Quarterly	Beginning

Applications

9. An investment of $100,000 today will brand advance quarterly payments of $4,000. If the annuity can earn 7.3% compounded semi-annually, how long will it take for the annuity to be depleted?
10. Amarjit wants to relieve up for a down payments on his offset abode. A typical starter home in his area sells for $250,000 and the banking concern requires a 10% down payment. If he starts making $300 month-stop contributions to an investment earning 4.75% compounded monthly, how long will it accept for Amarjit to have the necessary downward payment?
11. The neighbourhood grocery shop owned by Raoul needs $22,500 to upgrade its fixtures and coolers. If Raoul contributes $3,000 at the start of every quarter into a fund earning 5.4% compounded quarterly, how long will it take him to salvage up the needed funds for his store's upgrades?
12. Hullo-Tec Electronics is selling a 52" LG HDTV during a special "no sales tax" result for $1,995 with monthly payments of $100 including interest at 15% compounded semi-annually. How long will information technology have a consumer to pay off her new television?
13. Andre has stopped smoking. If he takes the $80 he saves each month and invests it into a fund earning half-dozen% compounded monthly, how long will it take for him to save $10,000?
14. How much longer volition a $500,000 investment fund earning 4.nine% compounded annually concluding if beginning-of-calendar month payments are $3,500 instead of $4,000?
15. Consider a $150,000 loan with month-end payments of $one,000. How much longer does it have to pay off the loan if the interest charge per unit is 6% compounded monthly instead of 5% compounded monthly?
16. In 1998, the Gillette Visitor launched the Mach three razor, having spent $750 1000000 in research and development along with an boosted $200 million in launching the production worldwide. Suppose the cost of borrowing was 10% compounded annually. If the forecast was to earn $eighty million in profits at the end of every quarter, how long did Gillette forecast it would take to pay back its investment in the Mach 3?

Challenge, Disquisitional Thinking, & Other Applications

17. Y'all make $250 month-stop contributions to your RRSP, which earns ix% compounded annually. a. How much less time will information technology take to achieve $100,000 if y'all increase your payments past x%? b. Which alternative requires less principal and past how much? (Assume all payments are equal.)
18. Most financial institutions tout the benefits of "topping up" your mortgage payments—that is, increasing from the required amount to whatsoever college amount. Assume a 25-year mortgage for $200,000 at a fixed rate of 5% compounded semi-annually.
    1. How many fewer payments does it take to pay off your mortgage if you increased your monthly payments by 10%?
    2. How much money is saved past "topping upwards" the payments? Assume that all payments are equal amounts in your calculations.
19. For an ordinary $250,000 loan with monthly payments of $2,000, do the post-obit calculations:
    1. How many payments are needed if the interest rate is half dozen% compounded annually? Semi-annually? Quarterly? Monthly?
    2. What is the total of the payments required under each alternative interest rate? The final payment amounts for each alternative are $193.nineteen, $ane,965.71, $1,950.13, and $ane,312.84, respectively.
    3. What tin y'all conclude from your various calculations in parts (a) and (b)?
20. For an ordinary $250,000 loan at vi% compounded monthly, practise the following calculations:
    1. How many payments are needed if the payments are $1,500 monthly? $4,500 quarterly? $9,000 semi-annually? $xviii,000 annually? (Notice that all of these options nominally pay the aforementioned corporeality per yr.)
    2. What is the total of the payments required under each alternative payment plan? The final payment amounts for each alternative are $371.24, $2,006.02, $420.61, and $8,300.30, respectively. c. What can y'all conclude from your various calculations in parts (a) and (b)?

eleven.half-dozen: Annuity Interest Rates

Mechanics

For questions one–eight, solve for both the nominal interest charge per unit indicated besides as the constructive interest rate.

Present Value	Future Value	Compounding Frequency	Annuity Payment	Term	Payment Timing
one. $0	$150,000	Monthly	$12,000 Annually	7 years	End
2. $500,000	$0	Quarterly	$8,000 Quarterly	21 years, 6 months	End
3. $0	$750,000	Semi-annually	$300 Monthly	33 years, 6 months	Beginning
iv. $100,000	$0	Monthly	$1,200 Monthly	10 years	Beginning
5. $25,000	$200,000	Semi-annually	$six,250 Semi-annually	eight years	End
6. $50,000	$25,000	Annually	$1,175 Monthly	2 years	Terminate
vii. $5,000	$35,000	Monthly	$2,650 Semi-annually	iv years, vi months	Beginning
8. $300,000	$150,000	Quarterly	$4,975 Quarterly	xiii years, iii months	Outset

Applications

9. Following his financial adviser'southward recommendations, Sanchez starts monthly contributions of $375 today to his RRSP. The programme is to take $240,000 in his RRSP after 20 years of monthly compounding. What nominal involvement rate does the financial adviser think Sanchez's RRSP volition be able to realize?
10. Helen'due south hubby recently passed away. The life insurance company is offering her a lump-sum payout of $250,000 today, or month-terminate payments of $1,585 for xx years.
    1. What monthly compounded and effective rate is the life insurance company using in its calculations?
    2. Helen thinks she can take the lump sum and invest it herself at 4.75% effectively. How much volition her monthly payment increment?
11. Francisco just inverse occupations. Unfortunately, he is not able to transfer his company pension with him to his new company. The administrators of the alimony plan offer him the selection of a lump-sum payout of $103,075 today or get-go-of-month payments of $535 for the adjacent 25 years. What semi-annually compounded rate of return are the pension administrators using in their calculations?
12. Under a wrongful dismissal adapt, a courtroom awarded a former employee $100,835.25 for finish-of-month wages of $4,500 for the past 21 months. What constructive interest rate is the courtroom using in the judgment?
13. A life insurance company recommends that its younger clients catechumen their term life insurance to permanent life insurance. One of the agents tells a client that if he invests $10,000 today along with $1,435 at the start of every quarter for the adjacent x years, he will own $100,000 of permanent life insurance. What semi-annually compounded rate of interest is beingness used?
14. Jake'southward Electronics wants to match a competitor'southward advertised payment program on an identical stereo system. If the system retails for $1,011.35 including all sales taxes and Jake wants to advertise six equal end-of-month payments of $174, what effective rate of interest does he need to charge?
15. The marketing managing director for Gilt'southward Gym offers members a ii-year membership for $650 in advance or first-of-month payments of $29. What monthly compounded interest is the marketing manager using in his pricing?
16. An investment today requires $ane,125.51 to purchase. In render, the investment pays out $30 after every six months for the next 20 years, along with an additional final lump-sum payout of $1,000. What semi-annually compounded interest rate is being earned on the investment?

Challenge, Critical Thinking, & Other Applications

17. A retail shop wants to offer its clients different 2-year ordinary payment plans on their product purchases. The marketing managing director understands the importance of odd-number pricing in these plans, where $499 is ameliorate than stating $500. On a typical $5,000 purchase, the marketing manager wants to offer payments of $229 monthly, $699 quarterly, or $one,399 semi-annually. The Competition Act of Canada requires total disclosure of the annual interest rate being charged on any plan. What interest rate must exist published for each plan?
18. When you lot purchase a automobile, a cash rebate is usually bachelor if yous finance the vehicle through your banking concern instead of the dealership; if you finance the vehicle through the dealership, y'all are not eligible for the greenbacks rebate. Assume you can purchase a vehicle for $24,960 and finance it for four years with month-end payments at 0% through the dealership. Alternatively, yous could go a loan from a bank and pay cash for your vehicle, which would entitle you to receive a $3,500 greenbacks rebate. What monthly compounded interest charge per unit would the banking concern have to charge to go far at the same monthly payment every bit the dealership alternative? What determination rule can y'all create from this adding?
19. On a $3,500 purchase, a company is thinking of offering a yr-long month-terminate payment program that requires payments of $299, $319, $334, or $349.
    1. If the goal of the program is to offer a competitive interest rate comparable to a depository financial institution credit card that averages 18% effectively, which payment plan should be chosen?
    2. If the goal of the programme is to offer a competitive interest rate comparable to a section store credit bill of fare that averages 28% effectively, which payment plan should exist chosen?
20. Margarite'south goal is to save up $100,000 after x years of monthly contributions into an investment annuity starting today. Depending on the level of hazard she chooses, her adviser tells her that under low-chance conditions she would need to contribute $645.nineteen, under medium-gamble conditions her contribution needs to be $523.32, and if she puts her money into high-run a risk investments she would demand $401.14 per calendar month. Based on the adviser'southward calculations, what are the effective interest rates on the depression-, medium-, and loftier-risk investments?

Review Exercises

Mechanics

1. Sangarwe will eolith $300 every quarter into an investment annuity earning 4.5% compounded quarterly for seven years. What is the departure in the amount of coin that she will have after 7 years if payments are fabricated at the beginning of the quarter instead of at the end?
2. Canseco wants to have enough money so that he could receive payments of $ane,500 every month for the next nine-and-ahalf years. If the annuity tin earn 6.one% compounded semi-annually, how much less money does he need if he takes his payments at the end of the month instead of at the starting time?
3. Kevin wants to save up $30,000 in an annuity earning 4.75% compounded annually so that he tin pay cash for a new car that he will purchase in three years' time. What is the difference in his monthly contributions if he starts today instead of one calendar month from now?
4. Brianne has a $21,000 loan beingness charged eight.4% compounded monthly. What are the month-terminate payments on her loan if the debt will be extinguished in 5 years?
5. Consider an investment of $225,000 earning v% annually. How long could it sustain annual withdrawals of $20,000 (including the smaller last payment) starting immediately?
6. The advertised month-end financing payments on a $28,757.72 car are $699 for a four-year term. What semi-annual and effective involvement rate is being used in the calculation?

Applications

7. Kubb Bakery estimates it volition demand $198,000 at a future point to aggrandize its production found. At the end of each calendar month, the profits of Kubb Baker average $xx,000, of which the possessor will commit 70% toward the expansion. If the savings annuity tin can earn 7.iii% compounded quarterly, how long will it take to enhance the necessary funds?
8. An investment fund has $7,500 in it today and is receiving contributions of $795 at the beginning of every quarter. If the fund tin can earn iii.8% compounded semi-annually for the first i-and-a-one-half years, followed by 4.35% compounded monthly for another ane-and-three-quarter years, what will be the maturity value of the fund?
9. A $17,475 Toyota Matrix is advertised with month-cease payments of $264.73 for six years. What monthly compounded charge per unit of return (rounded to 1 decimal) is being charged on the vehicle financing?
10. A variable charge per unit loan has a rest remaining of $17,000 later two years of fixed end-of-month payments of $655. If the monthly compounded interest charge per unit on the loan was 5.eight% for the first 10 months followed by 6.05% for 14 months, what was the initial amount of the loan?
11. Hank has already saved $68,000 in his RRSP. Suppose he needs to have $220,000 saved by the end of 10 years. What are his monthly payments starting today if the RRSP can earn 8.one% compounded annually?
12. Many companies keep a "slush fund" bachelor to embrace unexpected expenses. Suppose that a $15,000 fund earning 6.4% compounded semi-annually continues to receive month-end contributions of $1,000 for the adjacent five years, and that a withdrawal of $12,000 is made two-and-a-one-half years from today along with a second withdrawal of $23,000 four years from today. What is the maturity value of the fund?
13. Many consumers acquit a residuum each month on their credit cards and make minimal payments toward their debt. If a consumer owes $5,000 on a credit menu being charged 18.3% compounded daily interest, how long will information technology take him to pay off his debt with month-end payments of $100?
14. Y'all take a loan for $xx,000 on which you are charged 6% compounded quarterly. What payment amount at the stop of every vi months would reduce the loan to $15,000 later on ii years? What is the interest portion of the total payments made?

Challenge, Critical Thinking, & Other Applications

15. Stan and Kendra's children are currently four and two years old. When their older kid turns 18, they want to have saved upward plenty money and then that at the starting time of each twelvemonth they tin withdraw $20,000 for the starting time two years, $xl,000 for the side by side two years, and $twenty,000 for a final two years to subsidize their children's cost of postsecondary education. The annuity earns four.75% compounded semi-annually when paying out and 6.5% compounded monthly when they are contributing toward information technology. Starting today, what beginning-of-quarter payments must they deposit until their oldest reaches 18 years of age in order to accumulate the needed funds?
16. Karen is saving $1,500 at the end of every six months into an investment that earns nine.4% compounded monthly for the next 20 years. The maturity value will and then be rolled into an investment earning 5.85% compounded annually, from which she plans on withdrawing $23,800 at the commencement of each year. How long volition the annuity sustain the withdrawals (including the smaller final payment)?
17. In an effort to clear out last year'south vehicle inventory, Northside Ford advertises a vehicle at $46,500 with 0% financing for five years of end-of-month payments. Alternatively, consumers can pay greenbacks and receive a $half-dozen,000 rebate. What is the maximum monthly compounded interest rate that a bank could accuse that would outcome in equal or lower monthly payments?
18. Delaney is 18 years old and wants to sustain an annual income of $30,000 in today'southward dollars for 17 years at the end of every twelvemonth when she retires at age 65 (the amount will remain fixed once ready at age 65). If the annually compounded annuity tin can earn four.65% in retirement and nine.5% during contributions, how much does she demand to invest at the cease of every month? Assume the almanac rate of inflation is ii.vii%.
19. A mortgage can take up to 25 years to pay off. Taking a $250,000 home, calculate the month-terminate payment for 15-, 20-, and 25-year periods using semi-annually compounded interest rates of 4%, 5.5%, and 7% for each period. What do you observe from your calculations?
20. Being able to first an RRSP with a lump-sum investment tin reduce your end-of-month contributions. For any 35-yr term RRSP earning 8.7% compounded annually, summate the monthly contribution necessary to have a maturity value of $1,000,000 if the starting lump sums are $5,000, $ten,000, $15,000, and $xx,000. What do you observe from your calculations?

Source: https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Math_%28Olivier%29/11:_Compound_Interest_Annuities/11.E:_Compound_Interest-_Annuities_%28Exercises%29

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