# 12.E: Compound Interest- Special Applications Of Annuities (Exercises)

- Page ID
- 30941

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## 12.1: Deferred Annuities

### Mechanics

Accumulation Stage Payments | Payments Stage | |||||
---|---|---|---|---|---|---|

Present Value |
Nominal Interest Rate |
Period of Deferral |
Annuity Payments |
Annuity Payment Timing |
Annuity Term |
Nominal Interest Rate |

1. ? | 7.7% semiannually | 15½ years | $4,200 quarterly | End | 12 years | 5.9% semiannually |

2. $40,000 | 6.1% quarterly | ? | $908.56 monthly | Beginning | 10 years | 4.2% monthly |

3. $100,000 | 8.5% annually | 20 years | $20,000 semiannually | Beginning | ? | 3.9% quarterly |

4. $17,500 | 7.5% monthly | 14 years, 8 months | ? annually | End | 5 years | 5% annually |

5. $28,900 | 8.2% quarterly | ? | $17,018.98 semiannually | End | 7 years, 6 months | 2.8% annually |

6. $55,000 | 10.8% monthly | 13 years, 10 months | $9,850.80 quarterly | End | ? | 6.6% quarterly |

7. ? | 8.9% annually | 11 years | $1,500 monthly | Beginning | 20 years | 3% monthly |

8. $15,000 | 5.5% semiannually | 18 years | ? monthly | Beginning | 4 years | 4.1% semiannually |

### Applications

- What is the present value of a deferred annuity with a deferral period of 17 years at 6.7% compounded semi-annually followed by a 10-year annuity due paying $1,250 every month at 4.78% compounded semi-annually?
- Your objective is an annuity due paying $5,000 semi-annually for 5½ years at 4% compounded quarterly. How far in advance of this would you need to invest $20,000 at 6.82% compounded monthly? Assume 30 days in a month.
- If $38,000 is invested for 15 years at 9.4% compounded quarterly and then pays out $10,000 at the beginning of each year while earning 2.4% compounded annually, how far from today would the last payment occur?
- Jeff and Sarah want an ordinary annuity to pay their daughter $1,000 monthly for three years and nine months for the duration of her educational studies starting August 1, 2024. What lump-sum amount do they need to invest on August 1, 2014, if the deferred annuity can earn 6.6% compounded monthly during the accumulation stage and 4% compounded quarterly during the income payments stage?
- On July 13, 2011, Harriet invested $24,500 at 8.35% compounded semi-annually. Ten years later, she plans on withdrawing $5,000 at the end of every three months. If the income annuity can earn 4.5% compounded monthly, on what date will Harriet receive her last payment?
- At the age of 44, Parker just finished all the arrangements on his parents' estate. He is going to invest his $80,000 inheritance at 5.5% compounded quarterly until he retires at age 55, and then wants to receive month-end payments for the next 25 years. The income annuity is expected to earn 3.85% compounded annually. What are his monthly annuity payments during his retirement?
- Helga invested $34,000 on September 24, 2010, at 7.75% compounded monthly. She plans on using the money to fund an annuity due starting January 24, 2028, with the last payment being made on December 24, 2045. If the annuity due is expected to earn 5.8% compounded monthly, how much will she receive each month?

### Challenge, Critical Thinking, & Other Applications

- ING Direct just purchased a loan contract on its date of sale from a small retailer for $3,350.64 at a negotiated rate of 18% compounded monthly. If the contract calls for 24 month-end payments of $200 after a period of no payments, how long is the deferral period?
- Consider the following two investors saving into their RRSP earning 9% compounded annually throughout:
- Scully invests $275 at the end of every month from age 18 to age 35, then stops contributing until age 65 retirement.
- Mulder starts his RRSP later and invests $275 at the end of every month from age 35 to age 65.

- How much more money does the person with the higher balance have at age 65?
- How much money did each investor nominally put into the RRSP?
- What time value of money concept is being illustrated?

- Once Jason graduated college at age 22, he invested $350 into his RRSP at the beginning of every month until age 40. He then stopped his contributions and let the amount earn interest until today, when at age 62 he decided to retire. He wants his retirement money to last until age 85. If his account can earn 10.4% compounded quarterly before age 62 and 4.8% compounded annually after that, how much money can he expect to receive at the end of every quarter?
- Amber would like her RRSP earning 5.1% compounded semi-annually to pay her $2,500 at the end of every month for 20 years once she retires at age 65. What lump-sum amount at age 25 would Amber need to invest? Suppose she can get rates of 11.2% compounded annually for the first 35 years followed by 5.9% compounded annually until she needs the money at retirement.
- Compute the following scenarios using different interest rates of 6%, 8%, and 10% compounded annually throughout.
- What is the present value of a deferred annuity with a 10-year deferral period followed by a 10-year ordinary annuity with annual $10,000 payments?
- What is the annual annuity payment if a lump sum of $50,000 is invested for 10 years followed by a 10-year ordinary annuity?
- What is the term of the annuity if a lump sum of $50,000 is invested for 10 years followed by an ordinary annuity paying $20,000 annually?
- Discuss your observations from all of the above scenarios.

## 12.2: Constant-Growth Annuities

### Mechanics

**Solve each of the following constant growth annuities for **

**The future value****The present value**

Rate of Growth | Interest Rate | First Annuity Payment | Payment Timing | Term |
---|---|---|---|---|

1. 0.75% | 6% quarterly | $3,000 quarterly | End | 10 years |

2. 1.5% | 8% monthly | $5,000 semi-annually | End | 15 years |

3. 0.25% | 10% annually | $150 monthly | Beginning | 25 years, 11 months |

4. 4% | 8.5% annually | $7,500 annually | Beginning | 18 years |

5. 3% | 9.2% semi-annually | $4,000 annually | End | 40 years |

**Solve each of the following constant growth annuities for **

**The first annuity payment****The amount of the annuity payment corresponding to the payment number specified**

Rate of Growth | Value | Interest Rate | Payment Frequency | Payment Timing | Term | Payment Number |
---|---|---|---|---|---|---|

6. 0.15% | \(PV_{DUE}\) = $100,000 | 5.4% annually | monthly | Beginning | 10 years | 49 |

7. 3% | \(PV_{ORD}\) = $350,000 | 4.75% quarterly | annually | End | 15 years | 6 |

8. 0.8% | \(FV_{DUE}\) = $500,000 | 7.75% monthly | quarterly | Beginning | 25 years | 90 |

9. 1.75% | \(FV_{ORD}\) = $1,500,000 | 8.1% semi-annually | semi-annually | End | 45 years | last |

### Applications

- Yarianny wants to withdraw $25,000 annually starting today for the next 20 years and will increase the withdrawals by 3.5% each year. If the annuity can earn 6% compounded semi-annually, how much money needs to be invested in the fund today?
- Nikolay wants to make annual contributions to his RRSP for the next 25 years. He will increase each annual payment by 4.5%, and the RRSP can earn 9.3% compounded annually. If he wants to accumulate $250,000, what is the amount of his first payment today?
- A large municipality is saving up to build a new state-of-the-art publicly funded hockey arena to attract a National Hockey League (NHL) franchise. The municipality plans to invest $20 million at the end of the budget year and will increase contributions by 4% each year as its operating budget is expected to rise in future years. If the investment can earn 6.6% compounded annually, how much money rounded to the nearest dollar will the municipality have in 10 years?
- A five-year union contract calls for a company to provide $1 million in annual bonuses starting at the beginning of the first year and declining by 10% per year due to tough expected market conditions in the future. If the company wants to meet this commitment by putting a lump sum today into an investment earning 4.9% compounded quarterly, how much should it invest?
- Matthew bought a $200,000 annuity earning 5.75% compounded monthly. It will pay him at the end of every month for the next 20 years. The annuity is designed to make a large payment initially and then decrease by 0.4% per payment. a. What is the amount of his first annuity payment? b. Halfway through the annuity, what will his payment be? c. What is the amount of his last annuity payment?
- Cisco Systems plans a dividend of $1.50 per share at the end of the year and is expected to increase the dividend by 7% each year for the next 35 years. If an investor requires an interest rate of 12% compounded semi-annually on her investments, what is the value of a Cisco Systems share for that investor today based on the 35-year time horizon?
- An employee's pension fund is projected to have a value of $900,000 when he retires. An actuary determines that the employee has a life expectancy of 17 years after retirement. The pension fund can earn 4.3% annually, and the month-end payments will constantly increase by 0.25%.
- What is the amount of the first payment?
- What is the amount of the last payment?
- What is the total amount of the annuity payments over the entire term?

### Challenge, Critical Thinking, & Other Applications

- Harlen is 24 years old. His financial plan is to contribute $4,000 at the end of every year to his RRSP until age 60, growing each annual payment by 4.25%. At that point, he plans on retiring and using the money to fund 25 years of retirement. His month-end withdrawals will increase by 0.25% each month. The investment can earn 8.5% compounded annually and then 4.5% compounded annually in retirement.
- What is his total contribution to his RRSP?
- What is the amount of his first retirement payment?
- What are the total withdrawals he will make during retirement?

- The company accountant needs to record the current value of a pension liability for one of the company's employees. The employee just retired at age 67 and will receive $40,000 at the beginning of every year, increasing by 4% each year. Corporate actuaries have pegged the employee's expected life span at an estimated 23 years past retirement. The pension fund can earn 6.2% compounded semi-annually.
- What liability amount should the accountant record?
- Rounded to two decimals, what percentage of that amount is a result of the growth in the employee's payments?

- Recalculate the value of the Cisco Systems share in question 15 if the forecasted dividends are expected to continue into the infinite future. (Hint: use a very large value for N.) Comment on the difference obtained between your answer to question 15 and your answer to this question.
- Examine the impact of different growth rates on your RRSP.
- Start with your contributions to save up for retirement. In all cases, assume a starting contribution of $3,000 at the end of every year for a term of 40 years earning 9% compounded annually. Determine the future value of your RRSP with annual growth rates of 1%, 2%, 3%, 4%, and 5%. Also calculate the total payments made to the RRSP along with the interest earned. Comment on the results.
- Now examine your withdrawals after you have retired. In all cases, assume a starting withdrawal of $50,000 at the end of every year for a term of 16 years earning 4.8% compounded annually. Determine the present value required to fund your RRSP with annual growth rates of 1%, 2%, 3%, 4%, and 5%. Comment on the results.

## 12.3: Perpetuities

### Mechanics

**Solve each of the following perpetuities for the missing value (identified with a ?). **

Present Value | Interest Rate and Compounding Frequency | Payment Amount and Payment Frequency | Payment Timing |
---|---|---|---|

1. ? | 4.85% annually | $25,000 annually | End |

2. ? | 8.4% semi-annually | $20,000 semi-annually | Beginning |

3. $1,000,000 | 3.95% annually | ? annually | Beginning |

4. $2,400,000 | 4.4% quarterly | ? quarterly | End |

5. ? | 5.35% quarterly | $10,000 semi-annually | End |

6. $500,000 | 5.5% annually | ? semi-annually | Beginning |

7. $750,000 | 5% monthly | ? annually | End |

8. ? | 6.1% semi-annually | $7,500 quarterly | Beginning |

### Applications

- How much money is required today to fund a perpetuity earning 5.65% annually that needs to pay out $17,000 at the end of each year?
- If $2,500,000 was invested at 4.8% compounded semi-annually, what payment at the end of every six months could be sustained in perpetuity?
- The dean of the School of Business and Applied Arts at Red River College in Winnipeg wants to establish a scholarship program for the newly created finance major in its business administration stream. He wants to distribute five $2,000 scholarships annually starting immediately. How much money must he raise from college supporters if the perpetuity can earn 3.6% compounded monthly?
- Samson just won the grand prize of $50 million in the Lotto Max lottery. If he invests the money into a perpetuity fund earning 5.5% compounded annually, what monthly payment can he expect to receive starting today?
- An investor states that he would be willing to pay $25 for common shares to achieve his desired rate of return of 15% annually. What are the forecasted annual dividends starting one year from now?
- In 1910, an Aboriginal group signed a treaty with the British Columbia government in which they agreed to receive $1,542 annually in perpetuity. The government of today would like to pay off this debt. If prevailing interest rates are 3% compounded annually, how much should the Aboriginal group be willing to accept as payment in full for the perpetuity?
- The Coca-Cola Company is forecasted to have dividends of $0.50 per share quarterly for the next five years, followed by dividends of $0.70 per share quarterly in perpetuity. If an investor desires a 10% compounded annually rate of return, what amount would she be willing to pay per share?

### Challenge, Critical Thinking, & Other Applications

- Indigo's will states that $150,000 is to be set aside into a fund that will make annual payments to her grandson starting when he turns 18 years old. If Indigo dies when her grandson is six years old and the fund can earn 4.9% compounded quarterly, what annual payment will he receive in perpetuity?
- In 2009, the Canadian federal government committed $1.5 million in annual operating funding for the Canadian Museum for Human Rights in Winnipeg, which is scheduled to open in 2014. If the government had funded the museum by setting up an ordinary perpetuity in 2009 that could earn prevailing interest rates of 3% compounded annually, how much money would have been required?
- A rare example of perpetual bonds is the Consol bonds first issued by the British government in the 18th century. These perpetual bonds make end-of-quarter payments based on a semi-annual interest rate to the bondholders. If prevailing interest rates are 5%, what is the value of a bond today that pays $29.92 in total per year?
- Explore the impact of the interest rate on the principal required to fund an ordinary perpetuity. If the annual perpetuity payment is $10,000, calculate the principal required at annual interest rates of 2%, 3%, 4%, 5%, and 6%. Comment on your findings.
- Explore the impact of the interest rate on the ordinary perpetuity payment for a fixed principal. If the principal is $100,000, calculate the perpetuity payment at annual interest rates of 2%, 3%, 4%, 5%, and 6%. Comment on your findings.

## 12.4: Leases

### Mechanics

**For each of the following leases, determine the unknown variable (identified with a ?). **

Purchase Price | Down Payment | Lease Term (Years) | Nominal Interest Rate and Compounding Frequency | Lease Payment and Payment Frequency | Residual Value |
---|---|---|---|---|---|

1. ? | $5,000.00 | 3 | 5% monthly | $450.00 monthly | $11,500.00 |

2. $39,544.35 | $3,000.00 | 3 | 3.75% annually | $680.37 monthly | ? |

3. $45,885.00 | $4,500.00 | 4 | 5.9% monthly | ? quarterly | $9,750.00 |

4. $40,000.00 | $0.00 | 5 | ? annually | $851.95 monthly | $0.00 |

5. $85,000 | $15,000 | 1 | 9.99% monthly | $5,609.67 monthly | ? |

6. ? | $0.00 | 4 | 2.5% monthly | $26,500.00 quarterly | $0.00 |

7. $44,518.95 | $7,200.00 | 7 | ? quarterly | $2,627.17 semi-annually | $14,230 |

8. $175,000 | $10,000 | 6 | 13% semi-annually | ? quarterly | $5,000 |

### Applications

- A Ford F150 is on a 60-month lease at 5.99% compounded annually. Monthly payments are $397.95, and a $5,000 down payment was made. The residual value is $5,525.00. What was the purchase price of the truck?
- Pitney Bowes leases office equipment to other businesses. A client wants to lease $75,000 worth of equipment for 4½ years, at which point the equipment will have a residual value of $7,850. If Pitney Bowes requires a 15% compounded annually rate of return on its lease investments, what quarterly payment does it need to charge the client?
- Franklin is the office manager for Cargill Limited. He just signed a six-year leasing contract on some new production equipment. The terms of the lease require monthly payments of $32,385. If the alternative source of financing would have an interest rate of 7.3% compounded semi-annually, what capitalized lease liability should be recorded on the Cargill Limited balance sheet today?
- Harold is the production manager at Old Dutch Foods. He is considering leasing a new potato chip–making machine that will improve productivity. The terms of the 10-year lease require quarterly payments of $35,125 including interest at 6.99% compounded monthly. If the equipment is valued at $1,315,557.95 today, what is the estimated residual value of the equipment when the lease expires?
- Simi is considering leasing some computer equipment and peripherals on a two-year lease. The purchase price of the equipment is $40,674.35 with monthly payments of $1,668.86 including interest. The residual value is estimated to be 7.5% of its original value. What quarterly compounded interest rate, rounded to two decimals in percent format, is Simi being charged?
- Xerox’s outstanding sales agent, Sebastien, proudly boasts that he just completed a leasing arrangement on some new copier equipment for a three-year term at 17% compounded annually. It requires the client to make monthly payments of $2,375, and the equipment has a residual value of $2,500. What was the leasing price of the copier equipment?
- Unger Accounting Services leases its office and computer equipment. One year ago, it signed a three-year lease requiring quarterly payments of $1,600. Financing the equipment would have required a bank loan at 8.8% compounded monthly. How much has the capital leasing liability been reduced since the inception of the lease?
- Jerilyn operates a large farm in Alberta. She wants to lease a new harvester for her fields, one that has a purchase price of $120,000 and an estimated residual value of $50,000 after five years. The farm equipment dealer offers her a lease with a 6% annually compounded interest rate. What will her annual payments be?

### Challenge, Critical Thinking, & Other Applications

- A rent-to-own transaction is similar to a lease, where payments are required up front and the residual value equals zero. 1. What effective rate of interest (rounded to two decimals) is being charged for a two-year rent-to-own contract on a Panasonic camcorder with a purchase price of $379 requiring monthly payments of $79.88? What total amount of interest is paid? 2. If the individual borrowed money on a high-interest credit card instead with a rate of 29.99% compounded daily, what would be the monthly payments? What total amount of interest is paid?
- Fehrway Tours needs three extra touring vehicles for the next operating year because of a temporary increase in demand for its tour routes. Fehrway can buy the vehicles from Greyhound Canada for $175,000 each, or it can lease them for one year requiring monthly payments of $7,350 each based on a residual value of $110,000 each. If Fehrway purchases the vehicles, it requires financing at 6.6% compounded monthly and projects it could sell the vehicles for net revenue of $95,000 each after one year. Which option would you recommend that Fehrway pursue? How much better is the option in current dollars?
- Da-Young needs to record the total capital lease liability of her company. Based on the following outstanding capital leases, what amount should appear on her balance sheet?
- A five-year lease signed 2½ years ago requiring quarterly payments of $3,400 at 3.95% compounded annually and a residual value of $5,000.
- A seven-year lease signed 3¼ years ago requiring monthly payments of $895 at 6.8% compounded quarterly.
- A lease signed today for two years requiring semi-annual payments of $2,300 at 8.85% compounded annually with a residual value of $7,200.
- Some equipment leased with a purchase price of $200,000 three years ago on a six-year lease requiring semi-annual payments of $18,686.62 at 4.35% compounded semi-annually with a residual value of $21,000.

- Elena is shopping around for a one-year-old used Chevrolet Cobalt. The offers from three different used car dealerships are listed below. Which offer is financially best for Elena in current dollars? Calculate the present value and rank the three offers for Elena.
- Dealer #1: If she places $3,500 down, her four-year monthly lease payments will be $217.48 at 4.85% compounded monthly, and she will need to pay $3,629 at the end to buy her car outright.
- Dealer #2: If she places $3,000 down, her four-year monthly lease payments will be $237.53 at 4.65% compounded monthly and she will need to pay $3,400 at the end to buy her car outright.
- Dealer #3: If she places $2,000 down, her three-year monthly lease payments will be $259.50 at 4.5% compounded monthly and she will need to pay $6,000 at the end to buy her car outright.

## 12.5: Application - How To Purchase A Vehicle

**Because of the process required to make the vehicle purchase decision, this section offers only five exercises. You can solve these questions by the process presented using either formulas or the spreadsheet templates that accompany this text. For each of these questions, select the best method to acquire the vehicle based on the information provided. Identify the value of the payment under this best choice. Here are the options: **

- Lease from the Dealership
- Lease from the Bank
- Loan from the Dealership
- Loan from the Bank

### Mechanics

Note that all interest rates in the table are compounded annually. Note for the Banking Information that after calculating the dealer rates, you would go online and see if you can find something lower; the banking information below represents the results of your search.

Vehicle | Dealer Leasing Information | Dealer Purchase Information | Banking Information |
---|---|---|---|

1. Chevrolet Silverado |
3-year term 5.3% $295.00/month $17,810 residual value |
$2,000 cash rebate 5.79% |
Lease Rate = 7.15% Loan Rate = 8.8% Savings Rate = 3% |

2. Honda Civic |
4-year term 2.4% $236.46/month $8,520 residual value |
$2,500 cash rebate 7.9% |
Lease Rate = 8% Loan Rate = 10.9% Savings Rate = 2.25% |

3. Acura ZDX |
4-year term 2.4% $699.00/month $22,955.90 residual value |
$5,000 cash rebate 2.95% |
Lease Rate = 6.6% Loan Rate = 8.3% Savings Rate = 3.15% |

4. Dodge Grand Caravan |
5-year term 4.99% $141.00 biweekly $13,005 residual value |
$6,500 cash rebate 0% |
Lease Rate = 6.9% Loan Rate = 9.5% Savings Rate = 4.8% |

5. Toyota Matrix |
4-year term 0.9% $401.90/month $10,917.90 residual value |
$3,000 cash rebate 0.9% |
Lease Rate = 5.9% Loan Rate = 5.95% Savings Rate = 2.95% |

## 12.6: Application - Planning Your RRSP

**Due to the nature of these questions, five Application questions, two Challenge questions, and one Spreadsheet question are offered. **

### Applications

- When Criss turns 55, 35 years from now, he wants his RRSP savings to pay him the equivalent of $35,000 annually today with 2.4% annual inflation. He thinks his annual contributions can earn 9% compounded annually increasing by 4% per payment. Upon retirement, he wants to receive end-of-month payments for 30 years and estimates his account can earn 5.35% compounded annually. What is his first RRSP contribution payment at the end of the year?
- After graduating college at age 22, Kandahar immediately gained employment in the accounting field. He thinks that his career path will follow the “standard” path and he will retire at age 65. He feels he could live comfortably off of $30,000 in today’s dollars, with expected inflation of 2.1% annually. In retirement, he plans on receiving month-end payments for 13 years with 3.75% compounded monthly interest. Suppose his investment earns 8% compounded quarterly and he will increase his contributions by 0.9% per payment. What is his first quarterly RRSP contribution three months from now?
- Aisha and Richard are getting married, 33 years before they hit age 65. Neither has any RRSP savings to date. They figure that together in retirement they would like to earn the equivalent of $60,000 today plus 3% annual inflation. After retiring, they want to receive payments at the end of every three months for 15 years while the account earns an estimated 4.35% compounded semi-annually. If their RRSP can earn 8.25% compounded monthly and they plan on increasing contributions by 1.5% per interval, what first semi-annual payment is made today by each of them to achieve their joint goal?
- When Vanessa retires in 24 years, she wants her retirement fund to pay her the equivalent of $50,000 annually in today’s funds for 19 years while earning interest at a forecasted 5.15% compounded semi-annually. Her monthly payments start immediately upon the date of retirement. Current savings in her RRSP total $100,000, and historically she has been earning 7.35% compounded annually on the investment. She will make end-ofquarter contributions to her RRSP growing at 0.75% per payment. If the rate of inflation is 3% annually, what is the amount of the first contribution?
- Emma will make her first of 32 annual RRSP contributions today. Her financial adviser has indicated that she might earn approximately 6.5% compounded annually on her investment, and that she needs to increase each payment by 4%. After retiring at the beginning of the 33rd year from now, she wants month-end payments for 21 years that are the equivalent of $37,500 annually today while factoring in 2.55% annual inflation. Her retirement fund is expected to earn 4.05% compounded annually. She has current savings in her RRSP of $17,800. What is the amount of her first contribution?

### Challenge, Critical Thinking, & Other Applications

- Caitlin and Tu have three children to whom they will leave an inheritance of $100,000 each from the remaining balance in their retirement money. With 31 years until retirement, they have already managed to save $150,000 earning 6.9% compounded quarterly. They want to make month-end contributions increasing at 0.3% per contribution such that in retirement for 20 years they can receive on a month-end basis an annual income equivalent to $70,000 today. Inflation is expected to hover around 2.75% annually. The retirement fund is expected to earn 4.25% compounded annually. What is the first contribution required to achieve their desired goals?
- Wenli and Arjinder just celebrated the birth of their twins and are planning ahead for their education. In researching universities, they noted that the average tuition for an undergraduate student today is $5,138 and it is forecasted to rise by 4% annually into the foreseeable future. They plan that their education fund will make beginning-of-year payments for four years to each of their children when they turn 18 years old such that their education will be fully funded. Wenli and Arjinder will make semi-annual contributions rising by 2.5% each time. The education fund is expected to earn 7% compounded semi-annually, and when their kids turn 18 it will decrease to 5% compounded annually. What payment six months from now is required?
- Two key factors play an important role in RRSP calculations. These are the rate of inflation and the growth rate of the RRSP contributions. Use the information found in question 2 to answer the following:
- Using annual inflation rates of 2%, 3%, 4%, and 5%, calculate the RRSP contributions required under each scenario. What do you observe?
- Leave the inflation at 2.1% annually and recalculate the RRSP contributions required using growth rates of 0.5%, 0.7%, 0.9%, and 1.1%. What do you observe?

## Review Exercises

### Mechanics

- Four years from now, an annuity needs to pay out $1,000 at the end of every quarter for three years. Using an interest rate of 5% quarterly throughout, what amount of money must be invested today to fund the investment?
- Marnie wants to save up $250,000 to pay cash for a home purchase 15 years from now. If her investment can earn 6.1% compounded monthly and she intends to grow each payment by 0.25%, what will be her first monthly payment one month from now?
- Red Deer College wants to set up a scholarship for students in its business programs such that at the end of every year it could distribute a total of $50,000. If the perpetuity fund can earn 4.85% compounded semiannually, how much money will need to be raised to fund the scholarship?
- Carradine Industries needs to set a two-year quarterly lease price on some new equipment it is offering to its clients. If the company requires a 17% rate of return, what price should it charge on equipment valued at $44,750 with no expected residual value?
- Under a 48-month contract, Contact Marketing Inc. has been leasing $47,000 worth of servers and computer equipment for $1,292 per month for the past 23 months. What capitalized lease liability should be recorded on the balance sheet today if alternative financing could have been arranged for 9.4% compounded monthly?

### Applications

- Procter and Gamble shares are valued at $61.00 with perpetual year-end dividends of 3.1639%. What dividend payment would a holder of 750 shares receive in perpetuity assuming the share price and dividend rate remain unchanged?
- A Toyota RAV4 Limited 4WD V6 is advertised with low monthly lease payments of $488.86 for 48 months. The terms of the lease specify that the MSRP of the vehicle is $38,900 with a residual value of $17,502.80. A $6,250 down payment is required. What monthly compounded lease rate is Toyota charging?
- Jean-Rene wants to make a lump-sum deposit today such that at the end of every three months for the next five years he can receive a payment starting at $2,500 and increasing by 1% each time thereafter. At the end of the term, an additional lump-sum payment of $10,000 is required. If the annuity can earn 8.75% compounded semi-annually, what lump sum should he deposit today?
- A lump sum of $20,000 is invested at 6.25% compounded monthly for 18 years. It then pays out $2,500 at the end of every month while earning 5.05% compounded monthly. How many payments can the annuity sustain?
- The marketing manager for Infiniti places an advertisement for the new Infiniti QX56. Suppose the MSRP of the vehicle is $75,050 with a residual value after 48 months of $29,602. What monthly lease payment should she advertise if a $5,000 down payment is required and Infiniti Financial Services requires 5.9% compounded annually on all leases?
- Kraft Foods is planning for the replacement of some major production equipment. It will invest $1.475 million today at 5.95% compounded annually. The equipment purchase will be financed such that Kraft makes payments of $250,000 at the end of every quarter for two years at 6.65% compounded monthly. To the nearest day, how long does Kraft Foods need to wait to make the equipment purchase?
- The common shares of The Coca-Cola Company are forecast to pay $1.55 per share at the end of the next four years, and then $2.05 annually in perpetuity. If the market rate of return on such shares is 2.89%, what price should an investor be willing to pay today?
- A Hyundai dealership advertises a Hyundai Genesis for a low 60-month lease payment of $407.38 monthly at 0% interest with no down payment required. The residual value of the car is $16,316. A cash rebate of $4,000 is available. The lowest lease bank rate is 5.95% compounded monthly and savings accounts are paying 2.5% compounded annually. From the dealership, purchase financing is available for 1.9% compounded annually. Alternatively the consumer can place the car on their home equity line of credit at a rate of 3.5% compounded monthly. What is the lowest cost method to purchase this vehicle (lease or finance from either the dealer or the bank)?
- Klara is 25 years old and is planning for her retirement. She wants to consider both options of retiring at age 60 or at age 65, and has no savings into her RRSP yet. She figures she would like to earn in retirement the equivalent of $25,000 annually today. In retirement, she estimates the account would earn 4.35% compounded annually until age 82 before being depleted, all the while receiving end-of-month payments. She believes her RRSP can earn 8.4% compounded quarterly and she wants to grow her end-of-month contributions by 0.4% each. Supposing that annual inflation is expected to average 3%, what percentage larger will her first contribution be if she wants to retire at age 60 instead of age 65?

### Challenge, Critical Thinking, & Other Applications

- Sanchez is eligible to receive retirement benefits of $901.01 at age 65, but decides to start receiving them when he turns 60 instead. This incurs a 36% penalty on his benefits. By taking his retirement benefits at age 60, he will realize a monthly saving of $184.80 in retirement fund contributions. Both the benefits and contributions savings are expected to rise monthly by 0.15%. He will contribute the sum of these two amounts into his RRSP, which earns 7.35% compounded monthly until age 65, when he retires. In retirement, he will use the accumulated savings, which are expected to earn 4.5% compounded quarterly, to top up his retirement benefit payments to his pre-penalty amount. The monthly benefits are expected to rise monthly by 0.15%.
- Calculate the total of the monthly benefits and savings at age 60.
- Calculate the future value of the monthly savings and benefits at age 65.
- Calculate the amount of the initial penalty taken at age 60 and grow the penalty by 1.8% per year from age 60 to age 65.
- Take the amount of money in the fund at age 65 (from part (b)) along with the payments required to top up his retirement benefits to their regular level (from part (c)) to calculate the amount of money remaining in the fund at age 70.

- A Canadian college plans to implement a new business major in five years. To support the new program, it wants to offer 15 annual $2,500 scholarships at the beginning of each school year in perpetuity. If the scholarship fund can earn 4.65% compounded annually, what amount of money does the college need to raise today to fund the program?
- Luxmi wants to celebrate the birth of her first grandson by investing money for his future education. She estimates the annual cost of postsecondary education today to be $6,000 per year, and that the typical degree takes four years. She wants to invest a lump sum today that could sustain these payments with inflation starting 18 years from today. If annual inflation is 3.7%, and her investment can earn 6.6% compounded monthly, what amount does she need to invest today?
- AVCO Financial is under contract with a national retail chain to purchase its loan contracts on its date of sale. Under a special promotion, the retail chain allows a customer to defer her payments. If AVCO purchases the contract for $5,276.83 at its contractual rate of 21% compounded monthly and the consumer is required to make 30 month-end payments of $425, in how many months will AVCO receive its first payment?
- The Kenna Connect Marketing Group relies heavily on advanced technology to drive its business. A request for proposals on the next two-year computer equipment lease has resulted in four companies submitting bids:
- Company #1: Quarterly lease payments of $125,000 at 8.5% compounded quarterly with a residual value of $25,000.
- Company #2: Quarterly lease payments of $115,000 at 7.95% compounded quarterly with a residual value of $40,000 and a down payment of $80,000 required.
- Company #3: Monthly lease payments of $40,000 at 8.25% compounded monthly with a residual value of $35,000 and a down payment of $50,000 required.
- Company #4: Biweekly lease payments of $19,400 at 7.3% compounded monthly with a residual value of $42,000. Based on these bids, which company should Kenna Connect pursue the lease with and what is its cost in today's dollars?

- Jacques Cousteau just won the $10 million Powerball lottery. He has been offered the following choices on how to collect his winnings:
- Option 1: A one-time lump-sum payment of $4,289,771.59 today
- Option 2: A five-year deferred annuity followed by year-end payments of $400,000 for 25 years at 5% compounded annually throughout
- Option 3: A 35-year constant growth annuity with annual payments of $165,392.92 starting today and growing at 3% per year, earning interest at 5% compounded annually
- Option 4: Annual payments of $207,150 in perpetuity starting today earning 5% compounded annually Which option is the best financial choice? How much better in today's dollars is this option than the worst option?