9.E: Compound Interest- Working With Single Payments (Exercises)
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9.1: Compound Interest Fundamentals
For each of the following questions, round all percentages to four decimals where needed.
Mechanics
For questions 1–9, solve for the unknown variables (identified with a ?) based on the information provided.
| Nominal Interest Rate | Compounding Frequency | Periodic Interest Rate |
|---|---|---|
| 1. 7.2% | Monthly | ? |
| 2. 5.85% | Semi-annually | ? |
| 3. ? | Quarterly | 1.95% |
| 4. ? | Daily | 0.08% |
| 5. 9.2% | ? | 2.3% |
| 6. 5.5% | ? | \(0.458 \overline{3} \%\) |
| 7. ? | Annually | 14.375% |
| 8. 3.9% | Quarterly | ? |
| 9. 20.075% | ? | 0.055% |
Applications
- Calculate the periodic interest rate if the nominal interest rate is 7.75% compounded monthly.
- Calculate the compounding frequency for a nominal interest rate of 9.6% if the periodic interest rate is 0.8%.
- Calculate the nominal interest rate if the periodic interest rate is 2.0875% per quarter.
- Lori hears her banker state, “We will nominally charge you 10.68% on your loan, which works out to 0.89% of your principal every time we charge you interest.” What is her compounding frequency?
- You just received your monthly MasterCard statement and note at the bottom of the form that interest is charged at 19.5% compounded daily. What interest rate is charged to your credit card each day?
- Larry just purchased a new Ford Mustang from his local Ford dealer. His contract states that he will be charged interest at \(0.658 \overline{3} \%\) per month. What is his nominal interest rate?
Challenge, Critical Thinking, & Other Applications
- Betty just signed a contract for her business that will allow her to make interest-only payments for the next 12 months. If her interest rate is 12% compounded monthly and her outstanding principal is $5,000, how much will she pay in interest every month?
- Geoff is shopping around for a car loan. At three different websites he saw posted rates of 6.14% compounded semiannually, 3.06% compounded per six months, and 1.49% compounded every quarter. Which is the lowest nominal rate?
- After a period of three months, Alese saw one interest deposit of $176.40 for a principal of $9,800. What nominal rate of interest is she earning?
- Take a nominal interest rate of 10.4% and convert to its matching periodic interest rate if interest is compounded semiannually, quarterly, monthly, or daily.
- Take a periodic interest rate of 1.1% and convert to its matching nominal interest rate if the compounding period is per month, per quarter, or per six months.
9.2: Determining the Future Value
Mechanics
For questions 1–3, solve for the future value at the end of the term based on the information provided.
| Principal | Interest Rate | Term |
|---|---|---|
| 1. $7,500 | 8% compounded quarterly | 3 years |
| 2. $53,000 | 6% compounded monthly | 4 years, 3 months |
| 3. $19,000 | 5.75% compounded semi-annually | 6 years, 6 months |
For questions 4–6, solve for the future value at the end of the sequence of interest rate terms based on the information provided.
| Principal | Interest Rates and Term |
|---|---|
| 4. $3,750 | 4.75% compounded annually for 3 years; then 5.5% compounded semi-annually for 2 years |
| 5. $11,375 | 7.5% compounded monthly for 2 years, 9 months; then 8.25% compounded quarterly for 3 years, 3 months |
| 6. $24,500 | 4.1% compounded annually for 4 years; then 5.15% compounded quarterly for 1 year, 9 months; then 5.35% compounded monthly for 1 year, 3 months |
For questions 7–9, solve for the future value at the time period specified based on the information provided.
| Principal | Payments or Deposits | Interest Rates and Term | Find Future Balance At |
|---|---|---|---|
| 7. $2,300 loan | $750 payment at 9 months; $1,000 payment at 2 years | 9% compounded monthly for 9 months; then 8.75% compounded quarterly for 2.25 years | 3 years |
| 8. $4,000 investment | $1,500 deposit at 6 months; $1,500 deposit at 1½ years | 10% compounded semi-annually for 6 months; then 12% compounded quarterly for 1½ years | 2 years |
| 9. $3,000 loan | $300 payment at 4 months; $1,200 payment at 1 year, 9 months | 8% compounded monthly for 1 year, 9 months; then 9% compounded quarterly for 1 year, 9 months | 3½ years |
Applications
- You are planning a 16-day African safari to Rwanda to catch a rare glimpse of the 700 remaining mountain gorillas in the world. The estimated cost of this once-in-a-lifetime safari is $15,000 including the tour, permits, lodging, and airfare. Upon your graduation from college, your parents have promised you a $10,000 graduation gift. You intend to save this money for five years in a long-term investment earning 8.3% compounded semi-annually. If the cost of the trip will be about the same, will you have enough money five years from now to pay for your trip?
- Your investment of $9,000 that you started six years ago earned 7.3% compounded quarterly for the first 3¼ years, followed by 8.2% compounded monthly after that. How much interest has your investment earned so far?
- What is the maturity value of your $7,800 investment after three years if the interest rate was 5% compounded semiannually for the first two years, 6% compounded quarterly for the last year, and 2½ years after the initial investment you made a deposit of $1,200? How much interest is earned?
- Nirdosh borrowed $9,300 4¼ years ago at 6.35% compounded semi-annually. The interest rate changed to 6.5% compounded quarterly 1¾ years ago. What amount of money today is required to pay off this loan?
- Jason invested $10,000 into his RRSP when he turned 20 years old. Maria invested $10,000 into her RRSP when she turned 35 years old. If both earned 9% compounded semi-annually, what percentage more money (rounded to one decimal) will Jason have than Maria when they both turn 65 years old?
- Cristy borrowed $4,800 from a family friend 2½ years ago at 7% compounded annually for the first year and 8% compounded semi-annually thereafter. She made a payment 1½ years into the loan for $2,500. How much should Cristy pay today to clear her loan?
Challenge, Critical Thinking, & Other Applications
- A product manager wants to understand the impact of inflation on her gross profit margin. Inflation is expected to remain constant at 3.5% per year for the next five years. The price of the product is expected to remain unchanged at $99.99 over the five years and the current cost of the product today is $59.
- On a per-unit basis, in what dollar amount is the markup reduced after five years?
- What percent change does this represent in the gross profit margin?
- The Teachers’ Association just negotiated a four-year contract for its members, who will receive a 3.5% wage increase immediately followed by annual increases of 3.75%, 4.25%, and 4.1% on the anniversaries of the agreement. In the final year of the contract, how much more would the human resources manager for the school division need to budget for salaries if the average teacher currently earns $72,000 per year and the division employs 34 teachers?
- You invested $5,000 10 years ago and made two further deposits of $5,000 each after four years and eight years. Your investment earned 9.2% compounded quarterly for the first two years, 8.75% compounded monthly for the next six years, and 9.8% compounded semi-annually for the remaining years. As of today, how much interest has your investment earned?
- Suppose you placed $10,000 into each of the following investments. Rank the maturity values after five years from highest to lowest.
- 8% compounded annually for two years followed by 6% compounded semi-annually
- 8% compounded semi-annually for two years followed by 6% compounded annually
- 8% compounded monthly for two years followed by 6% compounded quarterly
- 8% compounded semi-annually for two years followed by 6% compounded monthly
- You made the following three investments:
- $8,000 into a five-year fixed rate investment earning 5.65% compounded quarterly.
- $6,500 into a five-year variable rate investment earning 4.83% compounded semi-annually for the first 2½ years and 6.5% compounded monthly for the remainder.
- $4,000 into a five-year variable rate investment earning 4.75% compounded monthly for the first two years and 5.9% compounded quarterly thereafter, with a $4,000 deposit made 21 months into the investment.
What is the total maturity value of all three investments after the five years, and how much interest is earned?
9.3: Determining the Present Value
Mechanics
For questions 1–3, solve for the present value at the beginning of the term based on the information provided.
| Maturity Value | Interest Rate | Term |
|---|---|---|
| 1. $14,000 | 9% compounded semi-annually | 14 years |
| 2. $202,754.18 | 7.85% compounded quarterly | 11 years |
| 3. $97,000 | 6% compounded monthly | 9 years, 3 months |
For questions 4–6, solve for the present value at the beginning of the sequence of interest rate terms based on the information provided.
| Maturity Value | Interest Rates (in order from the start of the transaction) and Term |
|---|---|
| 4. $45,839.05 | 4.5% compounded semi-annually for 4½ years; then 3.25% compounded annually for 4 years |
| 5. $7,223.83 | 8.05% compounded semi-annually for 2 years, 6 months; then 7.95% compounded quarterly for 1 year, 3 months; then 7.8% compounded monthly for 2 years, 9 months |
| 6. $28,995.95 | 18.75% compounded monthly for 4 years, 4 months; then 19.1% compounded daily for 3 years; then 18% compounded monthly for 1 year 7 months |
For questions 7–9, solve for the present value (today) based on the information provided.
| Maturity Amount and Date | Payments or Deposits (all times are from today) | Interest Rates (starting from today) and Term |
|---|---|---|
| 7. $7,500 due 4 years from today | $3,000 payment at 2 years | 8% annually for 1 year; then 6% semi-annually for 3 years |
| 8. $108,300 due 7 years from today | $50,000 payment at 3½ years; $25,000 payment at 6 years | 4.8% monthly for 3 years, 6 months; then 5.2% semi-annually for 3 years, 6 months |
| 9. $25,000 saved 5 years from today | $5,000 deposited at 2 years, 3 months; $5,000 deposited at 4 years | 9% quarterly for 1 year; Then 9¼% quarterly for 4 years |
Applications
10. Dovetail Industries needs to save $1,000,000 for new production machinery that it expects will be needed six years from today. If money can earn 8.35% compounded monthly, how much money should Dovetail invest today?
- A debt of $37,000 is owed 21 months from today. If prevailing interest rates are 6.55% compounded quarterly, what amount should the creditor be willing to accept today?
- Rene wants to invest a lump sum of money today to make a $35,000 down payment on a new home in five years. If he can place his money in an investment that will earn 4.53% compounded quarterly in the first two years followed by 4.76% compounded monthly for the remaining years, how much money does he need to invest today?
- Amadala owes Nik $3,000 and $4,000, due nine months and two years from today, respectively. If she wants to pay off both debts today, what amount should she pay if money can earn 6% quarterly in the first year and 5.75% monthly in the second year?
- Cheyenne just received her auto insurance bill. If she pays it in full today, she can deduct $15 off her total $950 premium. Alternatively, she can make two semi-annual payments of $475. If her money can earn 3% compounded monthly, which alternative should she pursue? How much money in today's dollars will she save in making that choice?
- Geoff is placing his money into a three-year investment that promises to pay him 8%, 8.25%, and 8.5% in consecutive years. All interest rates are compounded quarterly. If he plans to make a deposit to this investment in the amount of $15,000 18 months from now and his goal is to have $41,000, what amount does he need to invest today?
- In August 2004, Google Inc. made its initial stock offering. The value of the shares grew to $531.99 by July 2011. What was the original value of a share in August 2004 if the stock has grown at a rate of 26.8104% compounded monthly?
Challenge, Critical Thinking, & Other Applications
- In 2000 and 2009, Canada’s population was estimated at 31,496,800 and 33,487,208, respectively. In 2009, an estimated 74.9% of Canadians were known Internet users. If the historic annual rate of growth in Internet usage in Canada averaged 7.8568% per year, what percentage of the population in the year 2000 were Internet users?
- If a three-year and seven-month investment earned $8,879.17 of interest at 3.95% compounded monthly, what amount was originally placed into the investment?
- A lottery ticket advertises a $1 million prize. However, the fine print indicates that the winning amount will be paid out on the following schedule: $250,000 today, $250,000 one year from now, and $100,000 per year thereafter. If money can earn 9% compounded annually, what is the value of the prize today?
- Your company is selling some real estate and has received three potential offers:
- $520,000 today plus $500,000 one year from now.
- $200,000 today, $250,000 six months from now, and $600,000 15 months from now.
- $70,000 today, $200,000 in six months, then four quarterly payments of $200,000 starting six months later.
Prevailing interest rates are expected to be 6.75% compounded semi-annually in the next year, followed by 6.85% compounded quarterly afterwards. Rank the three offers from best to worst based on their values today.
9.4: Equivalent Payments
Mechanics
For questions 1–6, solve for the equivalent values (X) at the time period specified based on the information provided.
| Original Agreement | Proposed Equivalent Agreement | Prevailing Interest Rate |
|---|---|---|
| 1. $5,000 due today; $5,000 due in 3 years | $X due in 27 months | 6% compounded monthly |
| 2. $4,385 due 1 year ago; $6,000 due in 4 years | $X due in 2 years | 8.5% compounded quarterly |
| 3. $16,000 due today; $8,000 due in 10 months; $19,000 due in 33 months | $X due in 16 months | 6.65% compounded monthly |
| 4. $2,500 due in 3, 6, 9, and 12 months | $X due in 7 months | 8.88% compounded monthly |
| 5. $38,000 due 1½ years ago; $17,000 due ½ year ago; $45,000 due in 1 year | $X today | 9.5% compounded semi-annually |
| 6. $3,750 due 2½ years ago; $2,400 due in 3¼ years; $1,950 due in 5 years | $X due in 2 years | 10.75% compounded quarterly |
For questions 7–9, solve for the missing payment(s) on the loan based on the information provided.
| Loan Amount | Scheduled Payments | Loan Interest Rate |
|---|---|---|
| 7. $35,000 | $15,000 due in 6 months; $X due in 2 years | 5.95% compounded semi-annually |
| 8. $51,000 | $15,000 due in 3 months; $10,000 due in 1 year; $X due in 1½ years | 6.32% compounded quarterly |
| 9. $44,000 | $X due in 6 months; $2X due in 15 months | 5.55% compounded monthly |
Applications
- A winning lottery ticket offers the following two options:
- A single payment of $1,000,000 today or
- $250,000 today followed by annual payments of $300,000 for the next three years.
If money can earn 9% compounded annually, which option should the winner select? How much better is that option in current dollars?
- Darwin is a young entrepreneur trying to keep his business afloat. He has missed two payments to a creditor. The first was for $3,485 seven months ago and the second was for $5,320 last month. Darwin has had discussions with his creditor, who is willing to accept $4,000 one month from now and a second payment in full six months from now. If the agreed upon interest rate is 7.35% compounded monthly, what is the amount of the second payment?
- James is a debt collector. One of his clients has asked him to collect an outstanding debt from one of its customers. The customer has failed to pay three amounts: $1,600 18 months ago, $2,300 nine months ago, and $5,100 three months ago. In discussions with the customer, James finds she desires to clear up this situation and proposes a payment of $1,000 today, $4,000 nine months from now, and a final payment two years from now. The client normally charges 16.5% compounded quarterly on all outstanding debts. What is the amount of the third payment?
- Working in the accounting department, Ariel has noticed that a customer is having trouble paying its bills. The customer has missed a payment of $2,980 two years ago, $5,150 14 months ago, and $4,140 four months ago. The customer proposes making two payments instead. The first payment would be in six months and the second payment, one-quarter the size of the first payment, would be in 18 months. If the agreed-upon interest rate is 6.89% compounded monthly, what are the amounts of each payment?
- Seth works in the finance department of a large corporation. His company has the following debts to the same creditor: $98,000 due in 2 years; $203,000 due in 4¼ years, and $157,000 due in 6½ years. Seth wants to align these payments with the maturity dates of his company’s investments. He proposes to the creditor three payments due 1¾ years, 4½ years, and 5½ years from today. The second payment is to be twice the size of the first payment and the third payment is to be three-quarters the size of the second payment. If the creditor agrees to an interest rate of 7.25% compounded quarterly, calculate the amounts of each payment.
- A $30,000 loan at 4.9% compounded semi-annually is to be repaid with four equal semi-annual payments. The first payment is one year after the loan. Calculate the amount of each payment.
Challenge, Critical Thinking, & Other Applications
- The Ontario Labour Relations Board is reviewing a human resource complaint. At the time of filing, a construction worker indicated that her employer failed to pay her monthly wages of $3,400, $3,200, $3,600, and $3,475 starting four months prior. It has taken the Ontario Labour Relations Board nine months since the time of filing to gather the needed information and make a judgment in favour of the complainant. If the standard interest rate used in its judgment is 9% compounded monthly, what amount is awarded to the construction worker?
- Larry is a financial adviser helping a client save up $100,000 in five years’ time. The client has the financial means to pursue either of the following two alternatives:
- Make three equal deposits due today, in 2 years, and 4 years.
- Make four equal deposits due today, in 1 year, in 3 years, and 4½ years.
If Larry can invest the funds at 9% compounded semi-annually, which option is in the best interests of the client? In current dollars, how much better for the client is that recommended option?
- Four years ago Aminata borrowed $5,000 from Randal with interest at 8% compounded quarterly to be repaid one year from today. Two years ago Aminata borrowed another $2,500 from Randal at 6% compounded monthly to be repaid two years from today. Aminata would like to restructure the payments so that she can pay 15 months from today and 2½ years from today. The first payment is to be twice the size of the second payment. Randal accepts an interest rate of 6.27% compounded monthly on the proposed agreement. Calculate the amounts of each payment.
- Yi-Fang is the store owner of a franchise. In flipping through her records, she notices the following debts to the same supplier: $2,389.56 due eight months ago, $3,478.34 due six months ago, $1,694.32 due four months ago, $6,497.98 due two months ago, $4,611.03 due today, $5,784.39 due in two months, and $5,177.44 due in four months. She would like to clear all of these debts with a single payment next month. If the supplier charges 18.1% compounded monthly on overdue balances and provides a credit of 9% compounded monthly on early payments, calculate the amount of the payment.
- The City of Winnipeg has received two different offers to purchase a parcel of real estate in the southeast quadrant of the city. The Qualico Group has bid $12 million today, along with annual payments of $5 million for the first two years and $6 million in the subsequent two years. The Genstar Development Company has bid $7.5 million today, along with $8 million after one year and three subsequent annual payments of $7 million. If the prevailing interest rates are 8.75% compounded semi-annually, which offer should the City of Winnipeg accept? In current dollars, how much better is the highest bid?
9.5: Determining the Interest Rate
Mechanics
For questions 1–5, solve for the nominal interest rate compounded based on the information provided.
| Present Value (PV) | Future Value (FV) | Term | IY Compounding |
|---|---|---|---|
| 1. $101,000.00 | $191,981.42 | 10 years | monthly |
| 2. $78,500.00 | $144,935.53 | 7½ years | quarterly |
| 3. $59,860.48 | $78,500.00 | 5.5 years | semi-annually |
| 4. $31,837.58 | $1,000,000 | 40 years | annually |
| 5. $5,000.00 | $20,777.73 | 5 years | daily |
For questions 6–8, solve for the equal fixed nominal interest rate based on the information provided. Unless otherwise specified, assume each variable interest rate is for a one-year period.
| Variable Interest Rates | Fixed IY Compounding |
|---|---|
| 6. 6% compounded semi-annually for 3 years; then 6.5% compounded quarterly for 2 years | annually |
| 7. 3%, 3.5%, 4.25%, 5%, 6% all compounded quarterly | quarterly |
| 8. 2.4%, 3.6%, 4.2% all compounded monthly; then 6.25% and 8% compounded semi-annually | monthly |
Applications
- Your company paid an invoice five months late. If the original invoice was for $6,450 and the amount paid was $6,948.48, what monthly compounded interest rate is your supplier charging on late payments?
- In a civil lawsuit, a plaintiff was awarded damages of $15,000 plus $4,621.61 in interest for a period of 3¼ years. What quarterly compounded rate of interest was used in the settlement?
- Muriel just received $4,620.01 including $840.01 of interest as payment in full for a sum of money that was loaned 2 years and 11 months ago. What monthly compounded rate of interest was charged on the loan?
- Three years ago, Beverly made an investment that paid 4.95%, 5.55%, and 6.15% in each subsequent year. All interest rates are compounded monthly. What annually compounded fixed rate of return did she earn?
- Indiana just received a maturity value of $30,320.12 from a semi-annually compounded investment that paid 4%, 4.1%, 4.35%, 4.75%, and 5.5% in consecutive years. What amount of money did Indiana invest? What fixed quarterly compounded nominal interest rate is equivalent to the variable rate his investment earned?
- The five-year RRSP rate escalator program at TD Canada Trust offers semi-annually compounded interest rates of 1%, 2%, 3%, 4%, and 5%. A similar program at the Bank of Montreal posted rates of 1.1%, 1.55%, 2.75%, 3.25%, and 6.75%. Calculate the fixed semi-annual rates of return for each program and recommend where to invest in your RRSP. How much better is your recommended equivalent fixed rate?
Challenge, Critical Thinking, & Other Applications
- At what monthly compounded interest rate does it take five years for an investment to double?
- In 2003, a home in Winnipeg was purchased for $214,000. In 2011, the same home was appraised at $450,000. What annually compounded rate of growth does this reflect?
- On October 1, 1975, the minimum wage in Manitoba was $2.60 per hour. It rose to $10 per hour by October 1, 2011. What is the annually compounded growth rate for minimum wage in Manitoba during this period?
- Jean-Luc's first month's gross salary in June 1994 was $800. By June 2012 his monthly gross salary was $1,969.23 higher. What monthly compounded rate did his salary increase by over the period?
- Listed below are the historical populations (census metropolitan areas) of seven Canadian cities. Calculate and rank the annual growth rates from highest to lowest.
| City | 1996 | 2010 |
|---|---|---|
| Vancouver | 1,602,590 | 2,391,252 |
| Calgary | 754,033 | 1,242,624 |
| Regina | 191,692 | 215,138 |
| Winnipeg | 660,450 | 753,555 |
| Toronto | 3,898,833 | 5,741,419 |
| Montreal | 3,208,970 | 3,859,318 |
| Halifax | 332,518 | 403,188 |
- Listed below are several options for five-year investments posted on the same day. All rates are compounded quarterly. For each, calculate the equivalent fixed annually compounded rate of return and rank these rates from highest to lowest.
| Company | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
|---|---|---|---|---|---|
| Affinity Credit Union | 1% | 1.45% | 2.75% | 3.5% | 6% |
| MGI Financial | 1.85% | 2.6% | 3.3% | 3.41% | 3.75% |
| GIC Max | 2.1% | 2.75% | 3.4% | 3.75% | 3.8% |
| CIBC | 0.5% | 1.5% | 2% | 3.5% | 6.5% |
| Laurentian | 1.1% | 1.75% | 2.5% | 3.4% | 7% |
9.6: Equivalent and Effective Interest Rates
Mechanics
Convert the following nominal interest rates to their effective rates.
| Nominal Interest Rate |
|---|
| 1. 4.75% compounded quarterly |
| 2. 7.2% compounded monthly |
| 3. 3.95% compounded semi-annually |
Convert the following effective rates to their nominal rates.
| Effective Interest Rate | Nominal Interest Rate |
|---|---|
| 4. 10% | Semi-annually |
| 5. 12% | Monthly |
| 6. 8% | Quarterly |
Convert the following nominal interest rates to their equivalent rates.
| Nominal Interest Rate | Equivalent Rate |
|---|---|
| 7. 6% compounded monthly | Semi-annually |
| 8. 4.5% compounded quarterly | Monthly |
| 9. 8% compounded semi-annually | Quarterly |
| 10. 7.05% compounded monthly | Semi-annually |
Applications
- The HBC credit card has a nominal interest rate of 26.44669% compounded monthly. What effective rate is being charged?
- Your company's union just negotiated a salary increase of 0.5% every three months. What effective increase in your salary was negotiated?
- What is the effective rate on a credit card that charges interest of 0.049315% per day?
- RBC offers two different investment options to its clients. The first option is compounded monthly while the latter option is compounded quarterly. If RBC wants both options to have an effective rate of 3.9%, what nominal rates should it set for each option?
- Your competitor reported to its shareholders that growth in profits in the previous year averaged 0.1% per week (assume 52 weeks in a year). Your financial department shows your company’s profits have grown 0.45% per month. Compare your monthly profit growth to your competitor’s monthly profit growth. How much better is the higher growth?
- Louisa is shopping around for a loan. TD Canada Trust has offered her 8.3% compounded monthly, Conexus Credit Union has offered 8.34% compounded quarterly, and ING Direct has offered 8.45% compounded semi-annually. Rank the three offers and show calculations to support your answer.
Challenge, Critical Thinking, & Other Applications
- Your three-year monthly compounded investment just matured and you received $9,712.72, of which $2,212.72 was interest. What effective rate of interest did you earn?
- GenX Holdings received an invoice from its supplier that indicates the penalty on overdue invoices will be charged at a rate of 3.4% per month. What effective rate of interest is being charged on overdue invoices?
- The TD Emerald Visa card wants to increase its effective rate by 1%. If its current interest rate is 19.067014% compounded daily, what new daily compounded rate should it advertise?
- Five investors reported the following results. Rank the effective rates of return realized by each investor from highest to lowest and show outputs to support your work. Investor
9.7: Determining the Number of Compounds
Mechanics
Solve for the number of compounds involved in each transaction based on the information provided. Express your answer in a more common format.
| Present Value | Future Value | Nominal Interest Rate |
|---|---|---|
| 1. $68,000.00 | $89,032.97 | 4.91% compounded monthly |
| 2. $41,786.68 | $120,000.00 | 8.36% compounded quarterly |
| 3. $10,000.00 | $314,094.20 | 9% compounded annually |
| 4. $111,243.48 | $1,000,000.00 | 8.8% compounded semi-annually |
| 5. $25,000.00 | $125,000.00 | 5.85% compounded monthly |
| 6. $8,000.00 | $10,000.00 | 18% compounded daily |
| 7. $110,000.00 | $250,000.00 | 6.39% compounded weekly |
| 8. $500,000.00 | $2,225,500.00 | 7.1% compounded quarterly |
Applications
- You just took over another financial adviser's account. The client invested $15,500 at 6.92% compounded monthly and now has $24,980.58. How long has this client had the money invested?
- A debt of $7,500 is owed. Suppose prevailing interest rates are 4.9% compounded quarterly. How far in advance was the debt paid if the creditor accepted a payment of $6,721.58?
- Wayne was late in making a $3,500 payment to Dora. If Dora accepted a payment of $3,801.75 and charged 5.59% compounded semi-annually, how late was the payment?
- How long will it take $5,750 to become $10,000 at 6.25% compounded weekly?
- A friend of yours just won the 6/7 category on the Lotto Max (matching six out of seven numbers), and her share of the prize was $275,000. She wants to pay cash for a new home that sells for $360,000. If she can invest the money at 7.45% compounded semi-annually, how long will she have to wait to purchase the home assuming its sale price remains the same?
- Lakewood Properties anticipates that the City of Edmonton in the future will release some land for a development that costs $30 million. If Lakewood can invest $17.5 million today at 9.5% compounded monthly, how long will it take before it will have enough money to purchase the land?
- As marketing manager, you want to pursue a new product development for which you require $1 million for research. However, budgetary constraints mean you can only receive $850,000. If you take your budget and invest it at 8.7% compounded monthly, how long will it be before you can pursue the necessary research for the project?
Challenge, Critical Thinking, & Other Applications
- How long will it take money to double if it earns 8.2% compounded quarterly? First use the Rule of 72 to estimate the time, then compute the actual time. What is the difference between the two numbers (assume 182 days in a half year)?
- Your organization has a debt of $30,000 due in 13 months and $40,000 due in 27 months. If a single payment of $67,993.20 was made instead using an interest rate of 5.95% compounded monthly, when was the payment made?
- A $9,500 loan requires a payment of $5,000 after 1½ years and a final payment of $6,000. If the interest rate on the loan is 6.25% compounded monthly, when should the final payment be made?
- Your finance department has a policy of maturing its investments from longest held first to shortest when it needs the money for other purposes. Rank the following investments in the order that they should be matured as needed.
| Investment | Amount Invested | Current Valuation | Nominal Interest Rate |
|---|---|---|---|
| A | $50,000 | $75,549.62 | 7.94% compounded quarterly |
| B | $75,000 | $104,830.46 | 6.83% compounded monthly |
| C | $35,000 | $51,231.69 | 7.25% compounded semi-annually |
| D | $110,000 | $161,643.96 | 8.1% compounded quarterly |
- A loan requires five payments of $1,000 today, $1,500 due in 9 months, $3,000 due in 15 months, $2,500 due in 21 months, and $4,000 due in 33 months. Using an interest rate of 4.4% compounded monthly, a single payment of $11,950 was made to clear all debts. When was the single payment made
Review Exercises
Mechanics
- If you invest $10,000 at 7.74% compounded quarterly for 10 years, what is the maturity value?
- Ford Motor Company is considering an early retirement buyout package for some employees. The package involves paying out today's fair value of the employee's final year of salary. Shelby is due to retire in one year. Her salary is at the company maximum of $72,000. If prevailing interest rates are 6.75% compounded monthly, what buyout amount should Ford offer to Shelby today?
- Polo Park Bowling Lanes owes the same supplier $3,000 today and $2,500 one year from now. The owner proposed to pay both bills with a single payment four months from now. If interest rates are 8.1% compounded monthly, what amount should the supplier be willing to accept?
- What is the effective rate of interest on your credit card if you are being charged 24.5% compounded daily?
- Your 2½-year investment of $5,750 just matured for $6,364.09. What weekly compounded rate of interest did you earn?
- Vienna just paid $9,567.31 for an investment earning 5.26% compounded semi-annually that will mature for $25,000. What is the term of the investment?
Applications
- Merryweather's union just negotiated a new four-year contract with her employer. The terms of the contract provide for an immediate wage increase of 3.3%, followed by annual increases of 3.5%, 4.25%, and 2.75%. If she currently earns $61,500, what will her salary be in the final year of the contract?
- Bronco's four-year investment just matured at $26,178.21. If the investment earned semi-annually compounded interest rates of 4.5% and 4.75% in the first two years, followed by monthly compounded interest rates of 5% and 5.1% in the last two years, how much money did Bronco initially invest?
- Jay's Pharmacy owes the same creditor two debts of $6,000 due one year ago and $7,500 due in two years. Jay has proposed making two alternative payments of $10,000 due in three months and a final payment in 2½ years. If the creditor is agreeable to this proposal and wants an interest rate of 9% compounded quarterly, what is the amount of the final payment?
- Over a 10-year period, the Growth Fund of America had annual returns of 0.02%, 29.8%, 14.84%, 26.86%, 31.78%, 45.7%, 7.49%, −12.2%, −22.02%, and 32.9%. What fixed annually compounded rate of return did the fund realize over the 10 years?
- A sum of $84,100 was invested for 2½ years and matured at $101,268 using quarterly compounding. What quarterly compounded interest rate was realized? What is this effectively?
- A $10,000 loan at 8.15% compounded quarterly is to be repaid by two payments. The first payment is due in 9 months and the second payment, 1⅕ times the size of the first payment, is due in 33 months. Determine the amount of each payment.
- Exactly how long will it take for your money to quadruple at 6.54% compounded monthly?
- Jack is considering alternative three-year investment proposals from different banks for his $20,000 principal. RBC says that if he invests his money with them, he will have $23,841.78 using quarterly compounding. CIBC indicates that he would have $23,607.15 using monthly compounding. What nominal and effective rates are being offered by each bank?
Challenge, Critical Thinking, & Other Applications
- For the past four years, Darren has been saving up for a college fund for his oldest daughter. He deposited $5,000 initially, followed by annual deposits of $5,000, $4,000, $3,600, and $5,000. The money was initially invested at 5.75% compounded semi-annually for the first two years before increasing to 6% compounded quarterly for the balance of the investment. How much money is in the college fund today?
- Six months ago Old Dutch Foods purchased some new machinery for a new product line that they just developed. The supplier agreed to three payments on the machinery of $40,000 due today, $85,000 due in six months, and $75,000 due in 15 months. The new product line has not been as successful as initially planned, so Old Dutch Foods has proposed an alternative agreement involving three payments, each due at 3 months, 9 months, and 21 months. The second payment is to be double the first payment, and the last payment is to be double the second payment. If the supplier is agreeable to this and wants an interest rate of 8.55% compounded monthly, determine the payments required in the proposed agreement.
- Louisa owns a furniture store and decided to help a friend out by allowing him to purchase $5,000 of furniture using her credit at 6.25% compounded semi-annually. The furniture loan is to be repaid in four years. However, after 2½ years Louisa can no longer have the loan outstanding and needs the money. Avco Financial has agreed to purchase the maturity amount of this loan from Louisa using a discount rate of 17.1% compounded monthly. If Louisa proceeds with selling the loan contract to Avco Financial, what sum of money can she expect to receive?
- Franklin owes the following amounts to the same person: $16,000 due today, $11,500 due in 1¼ years, $17,000 due in 2¾ years, and $15,000 due in 4¼ years. He wants to make a single payment of $56,500 instead. Using an interest rate of 8% compounded quarterly, when should this payment be made?
- For each of the following investments, calculate the nominal rate of interest and convert to its effective rate of interest. Rank the investments from best to worst based on their effective rates.
| Investment | Present Value | Future Value | Compounded | Term |
|---|---|---|---|---|
| A | $31,000.00 | $39,190.09 | Weekly | 4 years, 8 weeks |
| B | $45,000.00 | $59,738.80 | Annually | 5 years |
| C | $37,310.00 | $43,534.31 | Quarterly | 2 years, 9 months |
| D | $16,175.00 | $19,701.35 | Semi-annually | 3 years, 6 months |
| E | $3,688.00 | $4,088.69 | Monthly | 1 year, 10 months |
- Three payments of $4,000 each are overdue to the same creditor by 1.5 years, 1 year, and 0.5 years. Three future payments of $4,000 each are due in 0.5 years, 1 year, and 1.5 years. Using an interest rate of 5.35% compounded quarterly, the debtor wants to make a single payment of $25,750. When should this payment be made?