# 13.E: Understanding Amortization and its Applications (Exercises)

- Page ID
- 29614

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## 13.1: Calculating Interest and Principal Components

### Mechanics

**For each of the following ordinary annuities, calculate the interest and principal portion of the payment indicated. **

Principal | Interest | Payment Frequency | Loan Term | Payment Number to Find |
---|---|---|---|---|

1. $5,000 | 8% compounded quarterly | Monthly | 3 years | 16 |

2. $45,000 | 7.65% compounded monthly | Monthly | 5 years | 23 |

3. $68,000 | 6.5% compounded semi-annually | Quarterly | 7 years | 19 |

4. $250,000 | 4.9% compounded annually | Semi-annually | 10 years | 17 |

**For each of the following ordinary annuities, calculate the total interest and principal portions for the series of payments indicated. **

Principal | Interest | Payment Frequency | Loan Term | Payment Series to Find (inclusive) |
---|---|---|---|---|

5. $20,000 | 5% compounded quarterly | Quarterly | 4 years | Year 1 |

6. $15,000 | 9% compounded monthly | Monthly | 4 years | Year 2 |

7. $39,000 | 6% compounded semi-annually | Quarterly | 5 years | Year 3 |

8. $50,000 | 7.5% compounded annually | Biweekly | 6 years | Year 4 |

9. $750,000 | 4% compounded monthly | Monthly | 10 years | 67 to 78 |

10. $500,000 | 6.25% compounded annually | Semi-annually | 8 years | 7 to 11 |

### Applications

- A $14,000 loan at 6% compounded monthly is repaid by monthly payments over four years.
- What is the size of the monthly payment?
- Calculate the principal portion of the 25th payment.
- Calculate the interest portion of the 33rd payment.
- Calculate the total interest paid in the second year.
- Calculate the principal portion of the payments in the third year.

- Quarterly payments are to be made against a $47,500 loan at 5.95% compounded annually with a six-year amortization.
- What is the size of the quarterly payment?
- Calculate the principal portion of the sixth payment.
- Calculate the interest portion of the 17th payment.
- Calculate how much the principal will be reduced in the fourth year.
- Calculate the total interest paid in the first year.

- A lump sum of $100,000 is placed into an investment annuity to make end-of-month payments for 20 years at 4% compounded semi-annually.
- What is the size of the monthly payment?
- Calculate the principal portion of the 203rd payment.
- Calculate the interest portion of the 76th payment.
- Calculate the total interest received in the fifth year.
- Calculate the principal portion of the payments made in the seventh year.

- For his son’s college education, Pat deposited $25,000 into an annuity earning 4.2% compounded quarterly. His son is to receive payments at the end of every quarter for five years.
- How much will his son receive each quarter?
- How much of the third payment is interest?
- How much of the payments made in the third year will come from the account’s principal?
- If his son finishes his education in four years instead of five and closes the account upon graduation, what total interest will he have received?

- Cathy and Bill just acquired a new Honda Odyssey Touring Edition minivan for $60,531.56 under the dealership’s purchase financing of 5.65% compounded annually for eight years.
- What are their monthly car payments?
- In the first year, what total amount of interest will they pay?
- In the fourth year, by how much will the principal be reduced?

### Challenge, Critical Thinking, & Other Applications

- Yangjing deposits $30,000 into an investment annuity for her daughter, who is currently living far away. The annuity is to earn 6.3% compounded semi-annually and make monthly payments starting today for the next five years. Calculate the interest portion of the payments made in the second year.
- At the age of 54, Hillary just finished all the arrangements on her parents' estate. She is going to invest her $75,000 inheritance at 6.25% compounded annually until she retires at age 65, and then she wants to receive month-end payments for the following 20 years. The income annuity is expected to earn 3.85% compounded annually.
- What are the principal and interest portions for the first payment of the income annuity?
- What is the portion of interest earned on the payments made in the second year of the income annuity?
- By what amount is the principal of the income annuity reduced in the fifth year?

- Art Industries just financed a $10,000 purchase at 5.9% compounded annually. It fixes the loan payment at $300 per month.
- How long will it take to pay the loan off?
- What are the interest and principal components of the 16th payment?
- For tax purposes, Art Industries needs to know the total interest paid for payments 7 through 18. Calculate the amount.

- Explore the impact of the term on the interest component of a loan. For a $200,000 loan at 5% compounded semiannually with monthly payments, calculate the following:
- Interest component for the entire loan for each term of 10, 15, 20, and 25 years.
- Between each increment of term in part (a), by what amount and what percentage did the annuity payment decrease?
- Between each increment of term in part (a), by what amount and percentage did the interest portion increase?
- Comment on your findings.

- Explore the impact of the interest rate on the interest component of a loan. For a $200,000 loan for 25 years with monthly payments, calculate the following:
- Interest component for the entire loan for each semi-annually compounded interest rate of 4%, 5%, 6%, 7%, and 8%.
- Between each increment of rate in part (a), by what amount and what percentage did the annuity payment decrease?
- Between each increment of rate in part (a), by what amount and what percentage did the interest portion increase?
- Comment on your findings.

## 13.2: Calculating the Final Payment

### Mechanics

**For each of the following ordinary annuities, calculate the final payment amount. **

Principal | Interest | Payment Frequency | Loan Term |
---|---|---|---|

1. $15,000 | 10% compounded quarterly | Quarterly | 3 years |

2. $85,000 | 6.75% compounded monthly | Monthly | 7 years |

3. $32,000 | 8.25% compounded annually | Annually | 10 years |

4. $250,000 | 5.9% compounded semi-annually | Monthly | 20 years |

**For each of the following ordinary annuities, calculate the final payment amount along with the total interest and principal portions for the series of payments indicated. **

Principal | Interest | Payment Frequency | Loan Term | Payment Series to Find (inclusive) |
---|---|---|---|---|

5. $21,000 | 8% compounded quarterly | Quarterly | 4 years | Year 4 |

6. $115,000 | 7% compounded semi-annually | Monthly | 10 years | Years 9 and 10 |

7. $13,750 | 9.5% compounded monthly | Semi-annually | 6 years | Years 5 and 6 |

8. $500,000 | 7.25% compounded semi-annually | Biweekly | 25 years | Year 25 |

9. $71,000 | 3.85% compounded quarterly | Annually | 15 years | Years 13 to 15 |

10. $47,500 | 10.25% compounded annually | Quarterly | 9 years | Year 9 |

### Applications

- A $28,250 loan at 9% compounded quarterly is repaid by monthly payments over five years. a. What is the amount of the final payment? b. Calculate the principal and interest portions of the payments in the final year.
- Semi-annual payments are to be made against a $97,500 loan at 7.5% compounded semi-annually with a 10-year amortization.
- What is the amount of the final payment?
- Calculate the principal and interest portions of the payments in the final two years.

- A $250,000 lump sum placed into an investment annuity is to make end-of-month payments for 17 years at 5% compounded annually.
- Calculate the principal and interest portions of the payments in the first five years.
- What is the amount of the final payment?
- Calculate the principal and interest portions of the payments in the last five years.

- A $65,000 trust fund is set up to make end-of-year payments for 15 years while earning 3.5% compounded quarterly.
- What is the amount of the final payment?
- Calculate the principal and interest portion of the payments in the final three years.

- Stuart and Shelley just purchased a new $65,871.88 Nissan Titan Crew Cab SL at 8.99% compounded monthly for a seven-year term.
- Calculate the principal and interest portions of the monthly payments in the first two years.
- What is the amount of the final monthly payment?
- Calculate the principal and interest portions of the payments in the last two years.

### Challenge, Critical Thinking, & Other Applications

- Mirabel Wholesale has a retail client that is struggling and wants to make instalments against its most recent invoice for $133,465.32. Mirabel works out a plan at 12.5% compounded monthly with beginning-of-month payments for two years.
- What will be the amount of the final payment?
- Calculate the principal and interest portions of the payments for the entire agreement.

- A new hotel built in Banff cost $36 million to build. The owner's financing arrangements allow for quarterly payments at 6% compounded semi-annually over the next 30 years. The first payment is to be made today.
- What is the amount of the final payment?
- Calculate the principal and interest portions of the payments in the final five years.

- Through a government arrangement, a new $110 million state-of-the-art baseball stadium will be constructed. Under terms of this arrangement, the owner of the baseball team will be charged 8.8% compounded annually and will be allowed to defer the payments for five years before making beginning-of-year payments for 40 years.
- What is the amount of the final payment?
- Calculate the principal and interest portion of the payments in the final five years.

- Wile E. Coyote owes the ACME Corporation $75,000 for various purchased goods. Wile agrees to make $1,000 payments at the end of every month at 10% compounded quarterly until the debt is repaid in full.
- What is the amount of the final payment?
- Calculate the principal and interest portion of the final six payments.

- Explore how the term affects the adjustment that needs to be made to a final payment. Consider a $450,000 loan at 7.5% compounded semi-annually with month-end payments.
- Calculate the final payment for each term of 10, 15, 20, 25, 30, and 35 years.
- Comment on the adjustments you made to the final payment based on your results.

## 13.3: Amortization Schedules

### Mechanics

**For each of the following ordinary annuities, create the complete amortization table and calculate the total interest. **

Principal | Interest | Payment Frequency | Loan Term |
---|---|---|---|

1. $27,500 | 6.8% compounded quarterly | Quarterly | 2 years |

2. $192,000 | 4.75% compounded semi-annually | Semi-annually | 4 years |

3. $1,854.25 | 8.99% compounded annually | Monthly | 6 years |

4. $425,000 | 5.9% compounded monthly | Annually | 7 years |

**For each of the following ordinary annuities, calculate the partial amortization schedule for the payment series indicated along with the total interest and principal portions. **

Principal | Interest | Payment Frequency | Loan Term | Partial Schedule Payments (inclusive) |
---|---|---|---|---|

5. $221,000 | 7.14% compounded quarterly | Quarterly | 15 years | Year 11 |

6. $109,900 | 3.8% compounded monthly | Monthly | 5 years | Payments 38 to 43 |

7. $450,000 | 7.5% compounded semi-annually | Monthly | 25 years | Payments 26 to 29 |

8. $500,000 | 9.5% compounded annually | Semi-annually | 12 years | Years 11 and 12 |

### Applications

**For questions 9 through 11, construct a complete amortization schedule and calculate the total interest. **

- A farmer purchased a John Deere combine for $369,930. The equipment dealership sets up a financing plan to allow for end-of-quarter payments for the next two years at 7.8% compounded monthly.
- Jennifer purchased an Aqua Shield Sunroom for $19,097. Terms of purchase require her to put $2,000 down, with the balance financed through six equal end-of-month payments at 11.2% compounded semi-annually.
- Jerry's Concrete installed a $13,544 concrete driveway for a client with no money down and four equal end-of-month installments at 4.9% compounded monthly.

**For questions 12 through 14, construct the partial amortization schedule indicated and calculate the total principal and interest portions represented by the partial schedule. **

- Some real estate is purchased for $850,000 with a 30-year amortization at 6.8% compounded semi-annually. Create a schedule for the end-of-month payments in the first half of the 14th year.
- Marcel had a new cedar fence installed around his oversized house lot for $22,900. The fence company allows him to finance the purchase with end-of-month instalments for one year at 7.39% compounded annually. Construct a schedule for the first two payments and the last four payments.
- Ron and Natasha had Oasis Leisure and Spa install an in-ground swimming pool for $51,000. The financing plan through the company allows for end-of-month payments for two years at 6.9% compounded quarterly. Ron and Natasha instruct Oasis to round their monthly payment upward to the next dollar amount evenly divisible by $500. Create a schedule for the first three payments, payments seven through nine, and the last three payments.

### Challenge, Critical Thinking, & Other Applications

- David leased an Acura CSX for $30,185 with no down payment on a 48-month lease at 2.9% compounded annually. The residual value is $11,516. Set up a partial amortization schedule for monthly payments 7 through 13. What is the total interest paid?
- Merryweather's $40,000 trust fund is set to mature and will make its first semi-annual payment to her today. The fund can earn 4.4% compounded annually during its five-year term. Construct a complete amortization schedule including the total interest earned.
- Construct a new complete amortization schedule for the farmer in question 9 if he has a prosperous third quarter and is able to put $35,000 toward his debt in addition to his regular third-quarter payment. How much interest in total would he save?
- Hillary acquired an antique bedroom set recovered from a European castle for $118,000. She will finance the purchase at 7.95% compounded annually through a plan allowing for payments of $18,000 at the end of every quarter.
- Create a complete amortization schedule and indicate her total interest paid.
- Recreate the complete amortization schedule if Hillary pays two additional top-up payments consisting of 10% of the principal remaining after her third payment as well as her fifth payment. What amount of interest does she save?

- Sinbad is an agent for RE/MAX. He purchased a Cadillac STS for his realty business so that he could drive clients to various homes. Through the bank, he financed the $85,595 vehicle at 5.95% compounded annually on a 24-month term. To file his taxes, he needs a complete amortization schedule with the total interest paid for the first 12 months and then for the next 12 months.
- Astrid just had a $450,000 custom home built by Hallmark Homes. She took out a 25-year amortization on her mortgage at 6.4% compounded semi-annually and locked the rate in for the first three years. Construct a partial amortization schedule for the first three years and calculate the total interest and principal portions.

## 13.4: Special Application - Mortgages

### Mechanics

**Determine the mortgage payment amount for each of the following mortgages. Assume all interest rates are fixed and compounded semi-annually. **

Principal | Interest Rate | Payment Frequency | Amortization Period | Mortgage Term (Years) |
---|---|---|---|---|

1. $693,482 | 5.49% | Monthly | 30 | 5 |

2. $405,551 | 6.5% | Bi-weekly | 25 | 3 |

3. $227,167 | 2.85 % | Weekly | 20 | 4 |

**Determine the mortgage payment amount upon renewal in the second term for each of the following mortgages. In all cases, assume the amortization period is reduced appropriately upon renewal and that all interest rates are fixed and compounded semi-annually. **

Original Principal | Amortization Period (Years) | First-Term Information | Second-Term Information |
---|---|---|---|

4. $434,693 | 30 | 4.5%; Monthly payments; 3-year term | 5.25%; Monthly payments; 2-year term |

5. $287,420 | 15 | 3.29%; Weekly payments; 1-year term | 6.15%; Monthly payments; 5-year term |

6. $318,222 | 25 | 3%; Biweekly payments 2-year term | 6.8%; Biweekly payments; 4-year term |

### Applications

- Jean-Claude wants to take out a mortgage in Montreal for $287,420 with a 20-year amortization. Prevailing interest rates are 6.2% compounded semi-annually. Calculate his monthly payment amount.
- Luc has noticed that mortgage interest rates are low and that he can take out a mortgage for 3.7% compounded semiannually. If he mortgages a home in Halifax for $255,818 amortized over 25 years, determine his biweekly payment amount.
- Five years ago, Asia purchased her $322,000 home in Edmonton with a 25-year amortization. In her first five-year term, she made monthly payments and was charged 4.89% compounded semi-annually. She will renew the mortgage on the same amortization timeline for another five-year term at 5.49% compounded semi-annually with monthly payments. Calculate the balance remaining after the first term and the new mortgage payment amount for the second term.

**For exercises 10 to 12, calculate the following: **

**Balance remaining after the first term****Interest and principal portions in the first term****New mortgage payment amount in the second term****Balance remaining after the second term**

- Three years ago, Phalatda took out a mortgage on her new home in Kelowna for $628,200 less a $100,000 down payment at 6.49% compounded semi-annually. She is making monthly payments over her three-year term based on a 30-year amortization. At renewal, she is able to obtain a new mortgage on a four-year term at 6.19% compounded semi-annually while continuing with monthly payments and the original amortization timeline.
- Dagny signed a 30-year amortization mortgage seven years ago on her home, which she purchased for $984,000 less a $150,000 down payment at 6.9% compounded monthly with monthly payments. She renews her mortgage on the same amortization schedule for another seven-year term at 7.15% compounded semi-annually with weekly payments.
- Two years ago, Eleonora started a $380,000 mortgage with a 25-year amortization at 4.15% compounded semi-annually with bimonthly payments. Upon renewal, she has decided to shorten the amortization period by five years and takes out a five-year term at 4.98% compounded annually with weekly payments.

**For exercises 13 and 14, calculate the following:**

**Balance remaining after all three terms****Total interest and total principal portion across all three terms**

- Ten years ago, Travis purchased his starter home with a mortgage principal of $180,700. He amortized the mortgage over 25 years and through his first five-year term he made monthly payments at 3.85% compounded semi-annually. Upon renewal, he took out another five-year term at 4.65% compounded semi-annually with monthly payments. Today, he renews his mortgage for another five-year term at 5.39% compounded semi-annually with monthly payments.
- Seven years ago, Aisha’s initial principal on her mortgage was $295,900. She set up a 20-year amortization and in her first four-year term of monthly payments her mortgage rate was 5.7% compounded semi-annually. Upon renewal, she took a further three-year term with monthly payments at a mortgage rate of 4.35% compounded semi-annually. Today, she renews the mortgage but decreases the amortization period by three years. She takes a five-year closed fixed rate mortgage of 6.39% compounded semi-annually with bimonthly payments.

### Challenge, Critical Thinking, & Other Applications

- Consider the following information about Jim and Carol, who are considering the purchase of a new home: Gross monthly total income $7,000 Estimated property taxes per month $400.00 Home heating costs per month $100.00 Monthly car payment $735.50 Monthly credit card debt payments $250.00 Monthly student debt payments $175.00 Planned Down Payment $15,000.00 Based on the affordability rules and a 25-year amortization, what is the highest price they could pay for their new home if interest rates are 5.19% compounded semi-annually?
- The Muswagons have signed a five-year closed variable rate $265,000 mortgage with a 25-year amortization and monthly payments. The initial interest rate was set at 4.5% compounded monthly. It increased by 0.75% after 14 months. Five months before the term expired, it then decreased by 0.25%. Calculate the balance remaining after the five-year term along with the total interest and principal portions paid.
- The Verhaeghes have signed a three-year closed fixed rate mortgage with a 20-year amortization and monthly payments. They negotiated an interest rate of 4.84% compounded semi-annually. The terms of the mortgage allow for the Verhaeghes to make a single top-up payment at any one point throughout the term. The mortgage principal was $323,000 and 18 months into the term they made one top-up payment of $20,000.
- What is the balance remaining at the end of the term?
- By what amount was the interest portion reduced by making the top-up payment?

- Assume a 15-year amortization on a $200,000 mortgage paid off through three consecutive five-year terms with monthly payments. Also assume that the same interest rate of 5.5% compounded semi-annually is maintained throughout the entire amortization. Construct a table showing the remaining balance, the principal portion, and the interest portion at the end of every term. (Hint: Don’t forget in the final term to make the appropriate final payment adjustment.)
- Fifteen years ago Clarissa’s initial principal on her mortgage was $408,650. She set up a 30-year amortization, and in her first 10-year term of monthly payments her mortgage rate was 7.7% compounded semi-annually. Upon renewal, she took a further five-year term with monthly payments at a mortgage rate of 5.69% compounded semi-annually. Today, she renews the mortgage but shortens the amortization period by five years when she sets up a three-year closed fixed rate mortgage of 3.45% compounded semi-annually with monthly payments. What principal will she borrow in her third term and what is the remaining balance at the end of the term? What total interest portion and principal portion will she have paid across all 18 years?
- Many people fail to understand how a mortgage rate increase would affect their mortgage renewal. The first decade of the new millennium has seen unprecedented low mortgage rates. As a result, many people purchased homes at the maximum mortgage that they could afford. At some point rates are going to rise again. When they do, many people may find themselves no longer able to afford their current homes since they were already maximized at the lower rate.
- Take a $350,000 mortgage with a five-year term and 25-year amortization using monthly payments. In the initial term, a rate of 3% compounded semi-annually was obtained. Upon renewal five years later on the same amortization schedule, explore the percent change in the monthly payment required if the semi-annually compounded interest rates had risen to each of 4%, 5%, and 6%. Comment on your findings.
- Assume that when the mortgage was started the homeowners had monthly taxes of $400 and home heating costs of $100 per month. If the mortgage was the maximum they could afford, according to the GDS ratio what was their annual gross income? Now assume that upon renewal the taxes and heating had increased to $450 and $120, respectively. At each of the new interest rates from part (a), by what percentage would the homeowners’ gross income need to rise over the same time frame for them to be able to continue to afford the mortgage? Comment on your findings.

## Review Exercises

### Mechanics

- Quinn placed $33,000 into a five-year ordinary investment annuity earning 7.75% compounded quarterly. He will be receiving quarterly payments. Calculate the principal and interest components of the sixth payment.
- Annanya took out a $42,500 ordinary loan at 6.6% compounded monthly with monthly payments over the six-year amortization period. Calculate the total principal and interest portions for the third year.
- Two years ago, Sumandeep invested $20,000 at 9.45% compounded monthly. She has been receiving end-of-month payments since, and the last payment will be today. Calculate the amount of the final payment.
- Hogwild Industries borrowed $75,000 to purchase some new equipment. The terms of the ordinary loan require quarterly payments for three years with an interest rate of 7.1% compounded semi-annually. Calculate the total interest and principal portions for the third year.
- Dr. Strong of Island Lakes Dental Centre acquired a new Panoramic X-ray machine for his practice. The $7,400 for the machine, borrowed at 8.8% compounded annually, is to be repaid in four end-of-quarter instalments. Develop a complete amortization schedule and total the interest paid.
- Dr. Miller acquired a new centrifuge machine from Liaoyang Longda Pharmaceutical Machinery Company (LLPMC) for his medical practice. He is to pay off the $60,341 through 20 month-end payments. LLPMC has set the interest rate on the loan at 9.5% compounded quarterly. Develop a partial amortization schedule for the third to sixth payments.
- Kerry, who is a pharmacist, just became a new franchisee for Shoppers Drug Mart. As part of her franchising agreement, her operation is to assume a $1.2 million mortgage to be financed over the next 15 years. She is to make payments after every six months. Head office will charge her a rate of 14.25% compounded annually. Determine the amount of her mortgage payment.
- Alibaba took out a 25-year amortization $273,875 mortgage five years ago at 4.85% compounded semi-annually and has been making monthly payments. He will renew the mortgage for a three-year term today at an interest rate of 6.1% compounded semi-annually on the same amortization schedule. What are his new monthly mortgage payments?

### Applications

- Monthly payments are to be made against an $850,000 loan at 7.15% compounded annually with a 15-year amortization.
- What is the size of the monthly payment?
- Calculate the principal portion of the 100th payment.
- Calculate the interest portion of the 50th payment.
- Calculate how much the principal will be reduced in the second year.
- Calculate the total interest paid in the fifth year.

- An investment annuity of $100,000 earning 4.5% compounded quarterly is to make payments at the end of every three months with a 10-year amortization.
- What is the size of the quarterly payment?
- Calculate the principal portion of the 20th payment.
- Calculate the interest portion of the 33rd payment.
- Calculate how much the principal will be reduced in the second year.
- Calculate the total interest paid in the seventh year.

- Presto Pizza just purchased a new $18,810 Chevrolet Aveo (including all taxes) as a delivery vehicle. The loan rate is 5.4% compounded annually for a seven-year amortization.
- What is the amount of the final monthly payment?
- Calculate the principal and interest portions of the payments in the first two years.
- Calculate the principal and interest portions of the payments in the last two years.

- A retail credit card allows its users to make purchases and pay off the debts through month-end payments equally spread over four months. The interest rate is 28.8% compounded annually on such purchases. A customer just placed $1,000 on the card and intends to use this plan.
- What is the amount of the final payment?
- Calculate the total interest incurred by using the payment plan.

- The marketing department at Allsports Industries has been assigned a lump sum of $5 million to use toward various research projects. The marketing research manager wants to invest the money at 6.4% compounded semi-annually and will withdraw $1 million at the end of every six months to fund ongoing research. Construct a complete amortization schedule and determine the total interest received.
- Horizon Insurance just leased $21,200 worth of new computer equipment for the next four years with no residual value expected. The lease rate is 7.9% compounded quarterly, and the company will make payments quarterly. Construct a partial amortization schedule and total the interest for the second year.
- Shelley Shearer Dance School took out a mortgage in Winnipeg for $500,000 on a five-year term with a 20-year amortization. The mortgage rate is 4.89% compounded semi-annually. Calculate the weekly mortgage payment amount.
- Four years ago, Katrina became a landlord and opened her new four-unit apartment housing unit with an initial mortgage at 6.83% compounded semi-annually in the amount of $971,000 less a $100,000 down payment. She amortized over 30 years and opted for monthly payments. Upon renewing her mortgage today, she is taking a two-year term at 5.1% compounded semi-annually while continuing with monthly payments and the original amortization timeline.
- Calculate the interest and principal portions in her first term.
- What is the balance remaining after the first term?
- What is the new mortgage payment amount in the second term?
- What is the balance remaining after the second term?

### Challenge, Critical Thinking, & Other Applications

- Five years ago, the Staples signed a closed fixed rate mortgage with a 25-year amortization and monthly payments. They negotiated an interest rate of 4.49% compounded semi-annually. The terms of the mortgage allow for the Staples to make a single top-up payment at any one point throughout the term. The mortgage principal was $179,000 and they made one top-up payment of $10,000 three years into the term. They are renewing the mortgage today for another five-year term but have reduced the amortization period by five years.
- What is the balance remaining at the end of the first term?
- By what amount was the interest portion of the first term reduced by making the top-up payment?
- Calculate the mortgage payment amount for the second term if the interest rate remains unchanged.

- The human resources department just received a $1 million gift from a former employee. The employee’s instructions are that the funds are to be invested with an amortization of 10 years and the employees of the company are to receive annual end-of-year Christmas bonuses from the fund. The company employs four managers, eight supervisors, and 20 workers. The annual bonuses are to be distributed such that a supervisor would receive twice as much as a worker, and a manager would receive twice as much as a supervisor.
- The fund is invested at 6.2% compounded semi-annually. Construct a complete amortization schedule for the investment.
- Except for the last year, how much can each worker, supervisor, and manager expect to receive as the annual Christmas bonus?

- Speedy Courier purchased a new Toyota Prius for $27,800 to add to its fleet of courier vehicles. It is being financed at 7.39% compounded monthly with monthly payments for six years. For each year of the vehicle loan, calculate the total interest and principal portions paid.
- B.O.S. Designer Candles Inc. has acquired some manufacturing space for $750,000 in an industrial section of the city. The owners intend to mortgage the property over a 15-year amortization period through successive five-year terms with monthly payments. The semi-annually compounded interest rate for the first term is 5%, and the owners foresee rates of 6% and 6.75% for the future terms. For each term, determine the mortgage payment amount, the total payments made, the total interest paid, and the total principal reduction.