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## 11.1: Fundamentals of Annuities

### Mechanics

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

- A debt of four payments of $500 due in 6 months, 12 months, 18 months, and 24 months.
- A debt of four quarterly payments in the amounts of $100, $200, $300, and $400.
- Contributions to an RRSP of $200 every month for the first year followed by $200 every quarter for the second year.
- Regular monthly deposits of $250 to an RRSP for five years, skipping one payment in the third year.

**For questions 5–8, determine the annuity type. **

Compounding Frequency | Payment Frequency | Payment Timing |
---|---|---|

5. Quarterly | Semi-annually | Beginning |

6. Annually | Annually | End |

7. Semi-annually | Semi-annually | Beginning |

8. Monthly | Quarterly | End |

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

- A $2,000 loan at 7% compounded quarterly is taken out today. Four quarterly payments of $522.07 are required. The first payment will be three months after the start of the loan.
- A new RRSP is set up with monthly contributions of $300 for five years earning 9% compounded semi-annually. The RRSP will have $22,695.85 when complete. The first payment is today.

### Applications

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

- Marie has decided to start saving for a down payment on her home. If she puts $1,000 every quarter for five years into a GIC earning 6% compounded monthly she will have $20,979.12. She will make her first deposit three months from now.
- 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.
- Brenda will lease a $25,000 car at 3.9% compounded monthly with monthly payments of $473.15 starting immediately. After three years she will still owe $10,000 on the vehicle.
- Steve takes out a two-year gym membership worth $500. The first of his monthly $22.41 payments is due at signing and includes interest at 8% compounded annually.
- Each year, Buhler Industries saves up $1 million to distribute in Christmas bonuses to its employees. To do so, at the end of every month the company invests $81,253.45 into an account earning 5.5% compounded monthly.

**For questions 16–20, assign the information in the timeline to the correct variables and determine the annuity type. Calculate the value of N.**

16.

17.

18.

19.

20.

## 11.2: Future Value Of Annuities

### Mechanics

**For questions 1–4, calculate the future value. **

Present Value | Interest Rate | Payments | Timing of Payment | Years |
---|---|---|---|---|

1. $0 | 7% quarterly | $2,000 quarterly | Beginning | 10 |

2. $0 | 9% monthly | $375 monthly | End | 20 |

3. $15,000 | 5.6% quarterly | $3,000 annually | End | 30 |

4. $38,000 | 8% semi-annually | $1,500 monthly | Beginning | 8 |

**For questions 5–8, calculate the future value.**

Present Value | Interest Rate | Payments | Timing of Payment |
---|---|---|---|

5. $0 | 6% annually for four years; then 7% semiannually for six years | $1,000 quarterly | End |

6. $0 | 12% quarterly for six-and-a-half years; then 11% semi-annually for three-and-a-half years | $100 monthly | Beginning |

7. $27,150 | 11% quarterly | $750 quarterly for five years; then $1,000 quarterly for 10 years | End |

8. $50,025 | 8% annually for three years; then 10% annually for seven years | $5,000 annually for three years; then $4,000 annually for seven years | Beginning |

### Applications

- Nikola is currently 47 years old and planning to retire at age 60. She has already saved $220,000 in her RRSP. If she continues to contribute $200 at the beginning of every month, how much money will be in her RRSP at retirement if it can earn 8.1% compounded monthly? No deposit is made the day she turns 60.
- You are a financial adviser. Your client is thinking of investing $600 at the end of every six months for the next six years with the invested funds earning 6.4% compounded semi-annually. Your client wants to know how much money she will have after six years. What do you tell your client?
- The Saskatchewan Roughriders started a rainy day savings fund three-and-a-half years ago to help pay for stadium improvements. At the beginning of every quarter the team has deposited $20,000 into the fund, which has been earning 4.85% compounded semi-annually. How much money is in the fund today?
- 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 $50,000 at the end of every quarter into a fund earning 6% quarterly, will it have enough money to build the warehouse two years from now? Show calculations to support your answer.
- The human resource 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 beginning of every biweekly pay period. Then for the next five years, she increased the deduction to $75. The company has been able to average 8.85% compounded monthly for the first seven years, and then 7.35% compounded monthly for the last three years. What amount has Margaret accumulated in her RRSP after 10 years? Assume there are 26 biweekly periods in a year.
- Joshua is opening up a Builder GIC that allows him to make regular contributions to his GIC throughout the term. He will initially deposit $10,000, then at the end of every month for the next five years he will make $100 contributions to his GIC. The annually compounded interest rates on the GIC in each year are 0.75%, 1.5%, 2.5%, 4.5%, and 7.25%. What is the maturity value of his GIC?
- How much more 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

- When Shayla turned five years old, her mother opened up a Registered Education Savings Plan (RESP) and started making $600 end-of-quarter contributions. The RESP earned 7.46% semi-annually. At the end of each year, Human Resources and Skills Development Canada (HRSDC) made an additional deposit under the Canada Education 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.
- Assume a 10-year ordinary annuity earning 10% compounded annually.
- If $5,000 is deposited annually, what is the maturity value?
- What is the maturity value if the deposits are doubled to $10,000? Compared to (a), what is the relationship between the size of the deposit and the maturity value, all other conditions being held equal?
- What is the maturity value if the $5,000 deposits are made semi-annually? Compared to (b), what is the relationship between the frequency of payments and the maturity value, all other conditions being held equal?

- Carlyle plans to make month-end contributions of $400 to his RRSP from age 20 to age 40. From age 40 to age 65, he plans to make no further contributions to his RRSP. The RRSP can earn 9% compounded annually from age 20 to age 60, and then 5% compounded annually from age 60 to age 65. Under this plan, what is the maturity value of his RRSP when he turns 65?
- To demonstrate the power of compound interest on an annuity, examine the principal and interest components of the maturity value in your RRSP after a certain time period. Suppose $200 is invested at the end of every month into an RRSP earning 8% compounded quarterly.
- Determine the maturity value, principal portion, and interest portion at 10, 20, 30, and 40 years. What do you observe?
- Change the interest rate to 9% quarterly and repeat (a). Comparing your answers to (a), what do you observe?

- Compare the following maturity values on these annuities due earning 9% compounded semi-annually:
- Payments of $1,000 quarterly for 40 years.
- Payments of $1,600 quarterly for 25 years.
- Payments of $4,000 quarterly for 10 years.

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

## 11.3: Present Value Of Annuities

### Mechanics

**For questions 1–3, calculate the amount of money that must be invested today for an individual to receive the future payments indicated and have the remaining balance at the end of the term.**

Future Value | Interest Rate | Payments | Timing of Payment | Years |
---|---|---|---|---|

1. $0 | 7% quarterly | $2,000 quarterly | Beginning | 10 |

2. $250,000 | 5.6% quarterly | $3,000 annually | End | 30 |

3. $380,000 | 8% semi-annually | $1,500 monthly | Beginning | 8 |

**For questions 4–6, calculate the amount of money that must be invested today for an individual to receive the future payments indicated and have the remaining balance at the end of the term. **

Future Value | Interest Rate | Payments | Payment Timing |
---|---|---|---|

4. $0 | 7% semi-annually for six years; then 6% annually for four years | $1,000 quarterly | End |

5. $327,150 | 11% quarterly | $1,000 quarterly for 10 years; then $750 quarterly for five years | End |

6. $150,025 | 10% annually for seven years; then 8% annually for three years | $4,000 annually for seven years; then $5,000 annually for three years | Beginning |

**For questions 7–8, calculate the balance owing and total interest paid over the time period indicated (from the start) for the following ordinary loans. **

Initial Loan Amount | Interest Rate | Payments | Balance Owing After |
---|---|---|---|

7. $35,000 | 8% quarterly | $853.59 monthly | Two years, six months |

8. $48,000 | 9% monthly | $865.23 monthly | Four years, eight months |

**For questions 9–10, calculate the proceeds of the sale for the following sales of ordinary loan contracts.**

Time Left on Loan on Date of Sale | Payments | Final Payment Amount | New Negotiated Interest Rate |
---|---|---|---|

9. Three-and-a-half years | $1,655.74 semi-annually | $1,655.69 | 14.85% monthly |

10. Four-and-a-quarter years | $1,126.96 monthly | $1,127.21 | 12.9% quarterly |

### Applications

- When Sinbad retires, he expects his RRSP to pay him $2,000 at the end of every month for 25 years. If his retirement annuity earns 3.8% compounded quarterly, how much money does he need to have in his RRSP when he retires?
- Sandy's parents would like to have an annuity pay her $500 at the beginning of every month from September 1, 2012, to April 1, 2017, to help with her university tuition and living expenses. On May 1, 2017, they would like to give her a graduation gift of $5,000. If the annuity can earn 6.15% compounded monthly, how much money must be in the account on September 1, 2012?
- 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 $3,000 at the end of every month for the next four years. Your human resource manager wants to set up an annuity to fund this obligation. If the proposed annuity can earn 5.7% compounded monthly for the first two-and-a-half years and then 6% compounded quarterly for the remaining one-and-a-half years, how much money should your company set aside today to meet its responsibilities?
- Working in the accounting department, Jaycee needs to accurately record 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 8.2% compounded quarterly. The payments at the end of every month have been $7,698.95. What amount should Jaycee record as the balance owing today? How much interest has been paid to date?
- Sleep Country Canada completed a sale of an entire mattress and box spring set to a client for $2,250 to be paid in 12 equal month-end instalments with no interest. If it immediately sells this contract on the date of issue to CitiFinancial at 12% compounded annually, what are the proceeds of the sale?
- Three years and two months ago, Mr. Magoo purchased a brand new Volkswagen Highline Jetta in Toronto for $32,854.75. He paid $5,000 as a down payment and financed the rest at 0.9% compounded monthly for six years. His payments have been $397.56 at the end of every month.
- What is the balance still owing on his vehicle today?
- If the dealership sells the loan contract today to a finance company at 9.9% compounded monthly, what are the proceeds of the sale? The last payment is for $397.85.

### Challenge, Critical Thinking, & Other Applications

- Lynne acquired a Sea Ray Sundancer boat and put $4,000 down. For the past two years, her end-of-month payments have been $1,049.01 including 9.32% compounded monthly. If she still owes $22,888.78 today, what was the purchase price of the boat?
- Gerald has been granted power of attorney and is now responsible for setting up his aging parents in a seniors’ home. The rent will be $2,490 at the beginning of every month for the first year, then increase by 5% the following year and 4% in the third year. Gerald wants to take money from his parents’ estate and set up an annuity to pay their monthly rent. If he can get an annuity that earns 3.75% semi-annually, how much money from his parents’ estate needs to be invested today to meet the rental payments over the next three years?
- Compare the amount of money that needs to be invested today to provide the required payments from the investment fund annuities earning 9% compounded semi-annually:
- Payments of $1,000 quarterly for 40 years.
- Payments of $1,600 quarterly for 25 years.
- Payments of $4,000 quarterly for 10 years. Note that in all three of these annuities the same total payout occurs. Explain your results and comment on your findings.

- HSBC Finance Canada is going to purchase the following ordinary loan contracts from the same company on the same date. In all cases, HSBC demands an interest rate of 18.9% compounded monthly on its purchases. What are the total proceeds of the sales?
- $734.56 quarterly payments with four years remaining in the term, and the final payment is $734.64.
- $1,612.46 semi-annual payments with six-and-a-half years remaining, and the final payment is $1,612.39.
- $549.98 monthly payments with five years and two months remaining, and the final 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 |
---|---|---|---|---|---|

1. \(FV_{DUE}\) = $25,000 | $0 | 5 years | Monthly | 8.25% | Monthly |

2. \(PV_{ORD}\) = $500,000 | $0 | 15 years | Quarterly | 5.9% | Semi-annually |

3. \(PV_{DUE}\) = $1,000,000 | $0 | 25 years | Monthly | 4.75% | Semi-annually |

4. \(FV_{ORD}\) =$1,500,000 | $0 | 35 years | Annually | 9% | Annually |

5. \(PV_{ORD}\) = $50,000 | \(FV\) = $5,000 | 6½ years | Semi-annually | 3.65% | Quarterly |

6. \(FV_{ORD}\) =$5,000,000 | \(PV\) = $450,000 | 4 years | Annually | 4.35% | Monthly |

7. \(FV_{DUE}\) = $1,000,000 | \(PV\) = $5,000 | 40 years | Monthly | 7.92% | Quarterly |

8. \(PV_{DUE}\) =$50,000 | \(FV\) =$10,000 | 5 years | Monthly | 5.5% | Annually |

### Applications

- To save approximately $30,000 for a down payment on a home four years from today, what amount needs to be invested at the end of every month at 4.5% compounded monthly?
- At age 60, Tiger has managed to save $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 5.4% compounded quarterly.
- If he plans on depleting the annuity, how much are his monthly payments?
- If he wants to have $50,000 left over at the end of the annuity, how much are his monthly payments?

- To purchase his new car, Scooby-Doo has obtained a six-year loan for $40,000 at 8.8% compounded semi-annually.
- Determine the monthly payments required on the loan.
- Calculate the balance owing and the total interest paid after three years of making payments toward the loan.
- Instead of (b), Scooby-Doo is considering reducing the balance owing to $15,000 after three years and paying off the loan in full at that time. What monthly payments are required?

- Gold's Gym wants to offer its clients a monthly payment option on its annual membership dues of $490. If the gym charges 7.75% compounded quarterly on its membership fee, what beginning-of-month payments should it advertise?
- A 20-year marketable bond can be purchased today for $13,402.90. It will make interest payments to the investor at the end of every six months, along with a $10,000 lump-sum payment to the investor at the end of the term. If prevailing interest rates are 6.85% compounded semi-annually, how much is each interest payment?
- Carling Industries needs to acquire some real estate to expand its operations. In negotiations with the Province of Nova Scotia, it will be allowed to purchase the $15 million parcel of land today and start making payments at the end of every six months for the next 10 years. If interest will be charged at 7.6% compounded semi-annually, what will be the required payments? (Round to the nearest dollar.)
- Sinclair does not believe in debt and will only pay cash for all purchases. He has already saved up $140,000 toward the purchase of a new home with an estimated cost of $300,000. Suppose his investments earn 7.5% compounded monthly. How much does he need to contribute at the beginning of each quarter if he wants to purchase his home in five years?
- A-One Courier Services needs to lease five vehicles for the next three 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 balance owing on each vehicle will be $10,300 at the end of the lease. Calculate the required lease payments.

### Challenge, Critical Thinking, & Other Applications

- A sales representative tells a production manager that if she purchases a new piece of machinery with a two-year life expectancy for $40,000 she will see a substantial reduction in operating costs. To purchase the machine, the production manager will need to obtain a two-year loan at 8% compounded quarterly. What is the least amount by which the monthly operating costs would need to be reduced for this purchase to make economic sense? Assume that operating costs are reduced at the beginning of each month.
- The Kowalskis’ only child is eight years old. They want to start saving into an RESP such that their son will be able to receive $5,000 at the end of every quarter for four years once he turns 18 and starts attending postsecondary school. When the annuity is paying out, it is forecast to earn 4% compounded monthly. While they make contributions at the end of every month to the RESP, it will earn 8% compounded semi-annually. Additionally, at the end of every year of contributions the government places a $500 grant into the RESP. What is the monthly contribution payment by the Kowalskis?
- Santana wants his retirement money to pay him $3,000 at the beginning of every month for 20 years. He expects the annuity to earn 6.15% compounded monthly during this time. If his RRSP can earn 10.25% compounded annually and he contributes for the next 30 years, how much money does he need to invest into his RRSP at the end of every month? He has already saved $15,000 to date.
- A lot of people fail to understand how interest rate changes affect their mortgages. Many think that if the interest rate on their mortgage rises from 5% to 6% their payments will rise by 1%. Assume a $100,000 mortgage with end-of-month payments for 25 years. Calculate the monthly mortgage payment at different semi-annually compounded interest rates of 4%, 5%, 6%, 7%, and 8%. What happens as the interest rate rises by 1% each time?

## 11.5: Number Of Annuity Payments

### Mechanics

**For questions 1–8, calculate the number of annuity payments required and express in a common date format. **

Present Value | Future Value | Interest Rate | Annuity Payments | Payment Timing |
---|---|---|---|---|

1. $0 | $100,000 | 6.35% Quarterly | $3,000 Quarterly | End |

2. $0 | $500,000 | 4.8% Monthly | $1,000 Monthly | Beginning |

3. $0 | $250,000 | 5.6% Semi-annually | $7,500 Annually | End |

4. $0 | $175,000 | 7% Annually | $2,500 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 | Beginning |

7. $50,000 | $0 | 7.2% Quarterly | $500 Monthly | End |

8. $1,000,000 | $0 | 9% Monthly | $40,000 Quarterly | Beginning |

### Applications

- An investment of $100,000 today will make 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?
- Amarjit wants to save up for a down payments on his first home. A typical starter home in his area sells for $250,000 and the bank requires a 10% down payment. If he starts making $300 month-end contributions to an investment earning 4.75% compounded monthly, how long will it take for Amarjit to have the necessary down payment?
- The neighbourhood grocery store 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 save up the needed funds for his store’s upgrades?
- Hi-Tec Electronics is selling a 52" LG HDTV during a special "no sales tax" event for $1,995 with monthly payments of $100 including interest at 15% compounded semi-annually. How long will it take a consumer to pay off her new television?
- Andre has stopped smoking. If he takes the $80 he saves each month and invests it into a fund earning 6% compounded monthly, how long will it take for him to save $10,000?
- How much longer will a $500,000 investment fund earning 4.9% compounded annually last if beginning-of-month payments are $3,500 instead of $4,000?
- Consider a $150,000 loan with month-end payments of $1,000. How much longer does it take to pay off the loan if the interest rate is 6% compounded monthly instead of 5% compounded monthly?
- In 1998, the Gillette Company launched the Mach 3 razor, having spent $750 million in research and development along with an additional $200 million in launching the product worldwide. Suppose the cost of borrowing was 10% compounded annually. If the forecast was to earn $80 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, Critical Thinking, & Other Applications

- You make $250 month-end contributions to your RRSP, which earns 9% compounded annually. a. How much less time will it take to reach $100,000 if you increase your payments by 10%? b. Which alternative requires less principal and by how much? (Assume all payments are equal.)
- Most financial institutions tout the benefits of "topping up" your mortgage payments—that is, increasing from the required amount to any higher amount. Assume a 25-year mortgage for $200,000 at a fixed rate of 5% compounded semi-annually.
- How many fewer payments does it take to pay off your mortgage if you increased your monthly payments by 10%?
- How much money is saved by "topping up" the payments? Assume that all payments are equal amounts in your calculations.

- For an ordinary $250,000 loan with monthly payments of $2,000, do the following calculations:
- How many payments are needed if the interest rate is 6% compounded annually? Semi-annually? Quarterly? Monthly?
- What is the total of the payments required under each alternative interest rate? The final payment amounts for each alternative are $193.19, $1,965.71, $1,950.13, and $1,312.84, respectively.
- What can you conclude from your various calculations in parts (a) and (b)?

- For an ordinary $250,000 loan at 6% compounded monthly, do the following calculations:
- How many payments are needed if the payments are $1,500 monthly? $4,500 quarterly? $9,000 semi-annually? $18,000 annually? (Notice that all of these options nominally pay the same amount per year.)
- 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 you conclude from your various calculations in parts (a) and (b)?

## 11.6: Annuity Interest Rates

### Mechanics

**For questions 1–8, solve for both the nominal interest rate indicated as well as the effective interest rate. **

Present Value | Future Value | Compounding Frequency | Annuity Payment | Term | Payment Timing |
---|---|---|---|---|---|

1. $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 |

4. $100,000 | $0 | Monthly | $1,200 Monthly | 10 years | Beginning |

5. $25,000 | $200,000 | Semi-annually | $6,250 Semi-annually | 8 years | End |

6. $50,000 | $25,000 | Annually | $1,175 Monthly | 2 years | End |

7. $5,000 | $35,000 | Monthly | $2,650 Semi-annually | 4 years, 6 months | Beginning |

8. $300,000 | $150,000 | Quarterly | $4,975 Quarterly | 13 years, 3 months | Beginning |

### Applications

- Following his financial adviser's recommendations, Sanchez starts monthly contributions of $375 today to his RRSP. The plan is to have $240,000 in his RRSP after 20 years of monthly compounding. What nominal interest rate does the financial adviser think Sanchez's RRSP will be able to realize?
- Helen's husband recently passed away. The life insurance company is offering her a lump-sum payout of $250,000 today, or month-end payments of $1,585 for 20 years.
- What monthly compounded and effective rate is the life insurance company using in its calculations?
- Helen thinks she can take the lump sum and invest it herself at 4.75% effectively. How much will her monthly payment increase?

- Francisco just changed occupations. Unfortunately, he is not able to transfer his company pension with him to his new company. The administrators of the pension plan offer him the choice of a lump-sum payout of $103,075 today or beginning-of-month payments of $535 for the next 25 years. What semi-annually compounded rate of return are the pension administrators using in their calculations?
- Under a wrongful dismissal suit, a court awarded a former employee $100,835.25 for end-of-month wages of $4,500 for the past 21 months. What effective interest rate is the court using in the judgment?
- A life insurance company recommends that its younger clients convert 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 beginning of every quarter for the next 10 years, he will own $100,000 of permanent life insurance. What semi-annually compounded rate of interest is being used?
- Jake's Electronics wants to match a competitor's advertised payment plan 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?
- The marketing manager for Gold's Gym offers members a two-year membership for $650 in advance or beginning-of-month payments of $29. What monthly compounded interest is the marketing manager using in his pricing?
- An investment today requires $1,125.51 to purchase. In return, 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

- A retail store wants to offer its clients different two-year ordinary payment plans on their product purchases. The marketing manager understands the importance of odd-number pricing in these plans, where $499 is better than stating $500. On a typical $5,000 purchase, the marketing manager wants to offer payments of $229 monthly, $699 quarterly, or $1,399 semi-annually. The Competition Act of Canada requires full disclosure of the annual interest rate being charged on any plan. What interest rate must be published for each plan?
- When you buy a car, a cash rebate is usually available if you finance the vehicle through your bank instead of the dealership; if you finance the vehicle through the dealership, you are not eligible for the cash 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, you could get a loan from a bank and pay cash for your vehicle, which would entitle you to receive a $3,500 cash rebate. What monthly compounded interest rate would the bank have to charge to arrive at the same monthly payment as the dealership alternative? What decision rule can you create from this calculation?
- On a $3,500 purchase, a company is thinking of offering a year-long month-end payment plan that requires payments of $299, $319, $334, or $349.
- If the goal of the plan is to offer a competitive interest rate comparable to a bank credit card that averages 18% effectively, which payment plan should be chosen?
- If the goal of the plan is to offer a competitive interest rate comparable to a department store credit card that averages 28% effectively, which payment plan should be chosen?

- Margarite's goal is to save up $100,000 after 10 years of monthly contributions into an investment annuity starting today. Depending on the level of risk she chooses, her adviser tells her that under low-risk conditions she would need to contribute $645.19, under medium-risk conditions her contribution needs to be $523.32, and if she puts her money into high-risk investments she would need $401.14 per month. Based on the adviser's calculations, what are the effective interest rates on the low-, medium-, and high-risk investments?

## Review Exercises

### Mechanics

- Sangarwe will deposit $300 every quarter into an investment annuity earning 4.5% compounded quarterly for seven years. What is the difference in the amount of money that she will have after seven years if payments are made at the beginning of the quarter instead of at the end?
- Canseco wants to have enough money so that he could receive payments of $1,500 every month for the next nine-and-ahalf years. If the annuity can earn 6.1% 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 beginning?
- Kevin wants to save up $30,000 in an annuity earning 4.75% compounded annually so that he can pay cash for a new car that he will buy in three years' time. What is the difference in his monthly contributions if he starts today instead of one month from now?
- Brianne has a $21,000 loan being charged 8.4% compounded monthly. What are the month-end payments on her loan if the debt will be extinguished in five years?
- Consider an investment of $225,000 earning 5% annually. How long could it sustain annual withdrawals of $20,000 (including the smaller final payment) starting immediately?
- The advertised month-end financing payments on a $28,757.72 car are $699 for a four-year term. What semi-annual and effective interest rate is being used in the calculation?

### Applications

- Kubb Bakery estimates it will need $198,000 at a future point to expand its production plant. At the end of each month, the profits of Kubb Bakery average $20,000, of which the owner will commit 70% toward the expansion. If the savings annuity can earn 7.3% compounded quarterly, how long will it take to raise the necessary funds?
- An investment fund has $7,500 in it today and is receiving contributions of $795 at the beginning of every quarter. If the fund can earn 3.8% compounded semi-annually for the first one-and-a-half years, followed by 4.35% compounded monthly for another one-and-three-quarter years, what will be the maturity value of the fund?
- A $17,475 Toyota Matrix is advertised with month-end payments of $264.73 for six years. What monthly compounded rate of return (rounded to one decimal) is being charged on the vehicle financing?
- A variable rate loan has a balance remaining of $17,000 after two years of fixed end-of-month payments of $655. If the monthly compounded interest rate on the loan was 5.8% for the first 10 months followed by 6.05% for 14 months, what was the initial amount of the loan?
- 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 8.1% compounded annually?
- Many companies keep a “slush fund” available to cover 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 next five years, and that a withdrawal of $12,000 is made two-and-a-half years from today along with a second withdrawal of $23,000 four years from today. What is the maturity value of the fund?
- Many consumers carry a balance each month on their credit cards and make minimal payments toward their debt. If a consumer owes $5,000 on a credit card being charged 18.3% compounded daily interest, how long will it take him to pay off his debt with month-end payments of $100?
- You have a loan for $20,000 on which you are charged 6% compounded quarterly. What payment amount at the end of every six months would reduce the loan to $15,000 after two years? What is the interest portion of the total payments made?

### Challenge, Critical Thinking, & Other Applications

- Stan and Kendra's children are currently four and two years old. When their older child turns 18, they want to have saved up enough money so that at the beginning of each year they can withdraw $20,000 for the first two years, $40,000 for the next two years, and $20,000 for a final two years to subsidize their children's cost of postsecondary education. The annuity earns 4.75% compounded semi-annually when paying out and 6.5% compounded monthly when they are contributing toward it. 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?
- Karen is saving $1,500 at the end of every six months into an investment that earns 9.4% compounded monthly for the next 20 years. The maturity value will then be rolled into an investment earning 5.85% compounded annually, from which she plans on withdrawing $23,800 at the beginning of each year. How long will the annuity sustain the withdrawals (including the smaller final payment)?
- In an effort to clear out last year's 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 cash and receive a $6,000 rebate. What is the maximum monthly compounded interest rate that a bank could charge that would result in equal or lower monthly payments?
- Delaney is 18 years old and wants to sustain an annual income of $30,000 in today's dollars for 17 years at the end of every year when she retires at age 65 (the amount will remain fixed once set at age 65). If the annually compounded annuity can earn 4.65% in retirement and 9.5% during contributions, how much does she need to invest at the end of every month? Assume the annual rate of inflation is 2.7%.
- A mortgage can take up to 25 years to pay off. Taking a $250,000 home, calculate the month-end 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?
- Being able to start an RRSP with a lump-sum investment can reduce your end-of-month contributions. For any 35-year term RRSP earning 8.7% compounded annually, calculate the monthly contribution necessary to have a maturity value of $1,000,000 if the starting lump sums are $5,000, $10,000, $15,000, and $20,000. What do you observe from your calculations?