8.E: Simple Interest- Working With Single Payments and Applications (Exercises)
- Page ID
- 36251
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)8.1: Principal, Rate, Time
For each of the following questions, round all money to two decimals and percentages to four decimals.
Mechanics
For questions 1–6, solve for the unknown variables (identified with a ?) based on the information provided.
Interest Amount | Principal or Present Value | Interest Rate | Time |
---|---|---|---|
1. ? | $130,000.00 | 8% | 9 months |
2. $4,000.00 | ? | 2% per month | 8 months |
3. $1,437.50 | $57,500.00 | ? per year | 4 months |
4. $103.13 | $1,250.00 | 9% | ? months |
5. ? | $42,250.00 | 1½% per month | ½ year |
6. $1,350.00 | ? | 6¾% | 3 months |
Applications
- Brynn borrowed $25,000 at 1% per month from a family friend to start her entrepreneurial venture on December 2, 2011. If she paid back the loan on June 16, 2012, how much simple interest did she pay?
- Alex took out a variable rate $20,000 loan on July 3 at prime + 4% when prime was set at 3%. The prime rate increased ½% on August 15. How much simple interest does Alex owe if the loan is paid back on September 29?
- How much simple interest is earned on $50,000 over 320 days if the interest rate is:
- 3%
- 6%
- 9%
- What relationship is evident between simple interest amounts and rates in your three answers above?
- If you placed $2,000 into an investment account earning 3% simple interest, how many months does it take for you to have $2,025 in your account?
- Cuthbert put $15,000 into a nine-month term deposit earning simple interest. After six months he decided to cash the investment in early, taking a penalty of 1% on his interest rate. If he received $393.75 of interest, what was the original interest rate before the penalty on his term deposit?
- If you want to earn $1,000 of simple interest at a rate of 7% in a span of five months, how much money must you invest?
- On March 3, when the posted prime rate was 4%, an investment of $10,000 was placed into an account earning prime + 2½%. Effective June 26, the prime rate rose by ¾%, and on October 4 it rose another ½%. On November 2, it declined by ¼%. If the money was withdrawn on December 15, how much simple interest did it earn?
- If $6,000 principal plus $132.90 of simple interest was withdrawn on August 14, 2011, from an investment earning 5.5% interest, on what day was the money invested?
- Jessica decided to invest her $11,000 in two back-to-back three-month term deposits. On the first three-month term, she earned $110 of interest. If she placed both the principal and the interest into the second three-month term deposit and earned $145.82 of interest, how much higher or lower was the interest rate on the second term deposit?
Challenge, Critical Thinking, & Other Applications
- The simple interest formula applies to any instance of constant unit growth. Assume that 200 units of product are sold today and you forecast a constant 20-unit sales increase every four weeks. What will be the sales in one year's time (assume a year has exactly 52 weeks)?
- Cade received an invoice for $5,200 dated August 13 with terms of 2/10, net 30 with a stated penalty of 2% simple interest per month. If Cade pays this invoice on December 13, what penalty amount is added to the invoice?
- Marrina is searching for the best way to invest her $10,000. One financial institution offers 4.25% on three-month term deposits and 4.5% on six-month term deposits. Marrina is considering either doing two back-to-back three-month term deposits or just taking the six-month deposit. It is almost certain that interest rates will rise by 0.5% before her first three-month term is up. She will place the simple interest and principal from the first three-month term deposit into the second three-month deposit. Which option should Marrina pursue? How much better is your recommended option?
- Goliath borrowed $20,000 on January 3 at prime + 4.5%. The prime rate changed throughout his term (of a non–leap year) as follows:
Date Effective | Prime Rate |
---|---|
January 3 | 2.25% |
February 17 | 2.5% |
April 30 | 3% |
June 15 | 2.75% |
September 15 | 3.25% |
- If he repaid his loan on December 24, what amount of simple interest should Goliath be charged?
- What equivalent fixed rate of simple interest did he get charged on his loan?
- Evaluate each of the following $10,000 investment alternatives and recommend the best alternative for investing any principal regardless of the actual amount. Assume in all cases that the principal and simple interest earned in prior terms are placed into subsequent investments.
Alternative 1 | Alternative 2 | Alternative 3 |
---|---|---|
6% for 1 year |
5% for 6 months, then 7% for 6 months |
5.25% for 3 months, then 5.75% for 3 months, then 6.25% for 3 months, then 6.75% for 3 months |
What percentage more interest is earned in the best alternative versus the worst alternative?
8.2: Moving Money Involving Simple Interest
For each of the following questions, round all money to two decimals and percentages to four decimals.
Mechanics
For questions 1–4, solve for the unknown variables (identified with a ?) based on the information provided.
Interest Amount | Principal / Present Value | Interest Rate | Time | Maturity / Future Value |
---|---|---|---|---|
1. ? | $16,775.00 | 0.5% per month | 6 months | ? |
2. ? | ? | 9% | 2 months | $61,915.00 |
3. $1,171.44 | ? | 5¾% | ? months | $23,394.44 |
4. $2,073.67 | $41,679.00 | ? | 227 days | ? |
Questions 5–8 involve either early or late payments. Calculate the following for each:
- The dollar amount of the early or late payment
- The amount of interest that forms the late penalty or early benefit
Original Payments | Proposed Payments | Interest Rate |
---|---|---|
5. $1,700 today | Seven months from today | 8% |
6. $3,800 two years from now | Today | 6.35% |
7. $950 two months from now; $1,000 seven months from now | A single payment one year from now | 4.95% |
8. $2,150 three months from now; $1,875 thirteen months from now | A single payment today | 11% |
Applications
- On January 23 of a non–leap year, a loan is taken out for $15,230 at 8.8% simple interest. What is the maturity value of the loan on October 23?
- An accountant needs to allocate the principal and simple interest on a loan payment into the appropriate ledgers. If the amount received was $10,267.21 for a loan that spanned April 14 to July 31 at 9.1%, how much was the principal and how much was the interest?
- Suppose Robin borrowed $3,600 on October 21 and repaid the loan on February 21 of the following year. What simple interest rate was charged if Robin repaid $3,694.63?
- How many weeks will it take $5,250 to grow to $5,586 at a simple interest rate of 10.4%? Assume 52 weeks in a year.
- Jayne needs to make three payments to Jade requiring $2,000 each 5 months, 10 months, and 15 months from today. She proposes instead making a single payment eight months from today. If Jade agrees to a simple interest rate of 9.5%, what amount should Jayne pay?
- Markus failed to make three payments of $2,500 scheduled one year ago, nine months ago, and six months ago. As his creditor has successfully sued Markus in small claims court, the judge orders him to pay his debts. If the court uses a simple interest rate of 1.5% per month, what amount should the judge order Markus to pay today?
Challenge, Critical Thinking, & Other Applications
- Over a period of nine months, $40.85 of simple interest is earned at 1.75% per quarter. Calculate both the principal and maturity value for the investment.
- The Home Depot is clearing out lawn mowers for $399.75 in an end-of-season sale that ends on September 30. Alternatively, you could wait until April 1 (assume February is in a non–leap year) and pay $419.95 on the same lawn mower. Assume your money can earn simple interest of 4.72% on a short-term investment.
- Which option should you choose?
- How much money will you have saved on September 30 by making that choice?
- Tula lent $15,000 to a business associate on August 12 at 6% simple interest. The loan was to be repaid on November 27; however, the business associate was unable to make the payment. As a consequence, they agreed that the associate could repay the debt on December 29 and that Tula would add $200 to the balance that was due on November 27. What rate of simple interest is Tula charging on the late payment?
- Merina is scheduled to make two loan payments to Bradford in the amount of $1,000 each, two months and nine months from now. Merina doesn't think she can make those payments and offers Bradford an alternative plan where she will pay $775 seven months from now and another payment seven months later. Bradford determines that 8.5% is a fair interest rate. What is the amount of the second payment?
- Alia took out three back-to-back short-term simple interest investments. On November 10, Alia had saved up a total of $12,986.75. Her interest rate for each investment is listed below:
Dates | Interest Rate |
---|---|
March 2 to June 19 | 4.13% |
June 19 to October 7 | 4.63% |
October 7 to November 10 | 4.43% |
How much did Alia originally invest on March 2? Assume that the principal and interest from a prior investment are both placed into the next investment.
- An accountant just received a payment from a client on January 26 in the amount of $14,500. The cheque stub states the payment is to be applied toward the client's three outstanding loans. The three loans were for $4,500 on May 12, $6,750 on July 11, and $8,000 on August 23. All loans are being charged simple interest of 6.85%. What is the remaining balance on the client's account?
8.3: Savings Accounts And Short-Term GICs
Mechanics
- A savings account at your local credit union holds a balance of $5,894 for the entire month of September. The posted simple interest rate is 1.35%. Calculate the amount of interest earned for the month (30 days).
- If you place $25,500 into an 80-day short-term GIC at TD Canada Trust earning 0.55% simple interest, how much will you receive when the investment matures?
- In November, the opening balance on a savings account was $12,345. A deposit of $3,000 was made to the account on November 19, and a withdrawal of $3,345 was made from the account on November 8. If simple interest is paid at 0.95% based on the daily closing balance, how much interest for the month of November is deposited to the account on December 1?
- A 320-day short-term GIC earns 0.78% simple interest. If $4,500 is invested into the GIC, what is the maturity value and how much interest is earned?
Applications
- In February of a leap year, the opening balance on your savings account was $3,553. You made two deposits of $2,000 each on February 5 and February 21. You made a withdrawal of $3,500 on February 10, and another withdrawal of $750 on February 17. If simple interest is calculated on the daily closing balance at a rate of 1.45%, how much interest do you earn for the month of February?
- An investor places $30,500 into a short-term 120-day GIC at the Bank of Montreal earning 0.5% simple interest. The maturity value is then rolled into another short-term 181-day GIC earning 0.57% simple interest. Calculate the final maturity value.
- The Mennonite Savings and Credit Union posts the following rate structure for its Daily Interest Savings Account. The entire balance is subject to the highest posted rate, and interest is calculated daily based on the closing balance.
Balance | Interest Rate |
---|---|
$0–$10,000.00 | 0.1% |
$10,000.01–$25,000.00 | 0.21% |
$25,000.01–$50,000.00 | 0.34% |
$50,000.01–$100,000.00 | 0.85% |
$100,000.01 and up | 1.02% |
The opening balance in February of a non–leap year was $47,335. The transactions on this account were a deposit of $60,000 on February 8, a withdrawal of $86,000 on February 15, and a deposit of $34,000 on February 24. What interest for the month of February will be deposited to the account on March 1?
- An investor with $75,000 is weighing options between a 200-day GIC or two back-to-back 100-day GICs. The 200-day GIC has a posted simple interest rate of 1.4%. The 100-day GICs have a posted simple interest rate of 1.35%. The maturity value of the first 100-day GIC would be reinvested in the second 100-day GIC (assume the same interest rate upon renewal). Which alternative is best and by how much?
- Canadian Western Bank offers a Summit Savings Account with posted interest rates as indicated in the table below. Only each tier is subject to the posted rate, and interest is calculated daily based on the closing balance.
Balance | Interest Rate |
---|---|
$0–$5,000.00 | 0% |
$5,000.01–$1,000,000.00 | 1.05% |
$1,000,000.01 and up | 0.80% |
December's opening balance was $550,000. Two deposits in the amount of $600,000 each were made on December 3 and December 21. Two withdrawals in the amount of $400,000 and $300,000 were made on December 13 and December 24, respectively. What interest for the month of December will be deposited to the account on January 1?
- Sun Life Financial Trust offers a 360-day short-term GIC at 0.65%. It also offers a 120-day short-term GIC at 0.58%. You are considering either the 360-day GIC or three consecutive 120-day GICs. For the 120-day GICs, the entire maturity value would be "rolled over" into the next GIC. Assume that the posted rate increases by 0.1% upon each renewal. If you have $115,000 to invest, which option should you pursue and how much more interest will it earn?
Challenge, Critical Thinking, & Other Applications
- The Regular Savings Account at Steinbach Credit Union posts the following tiered rate structure. Interest is calculated on the entire balance for any given tier.
Balance | Interest Rate |
---|---|
$0–$100,000.00 | 1.75% |
$100,000.01–$250,000.00 | 1.85% |
$250,000.01 and up | 1.95% |
For any given month, simple interest is calculated on the lowest monthly closing balance, but the interest is not deposited until January 1 of the following year. On January 1, 2014, the opening balance on an account was $75,000. The following transactions took place throughout the year:
Date | Activity | Amount |
---|---|---|
February 14 | Deposit | $133,000 |
March 30 | Deposit | $42,000 |
June 15 | Withdrawal | $110,000 |
August 1 | Deposit | $212,000 |
October 29 | Withdrawal | $300,000 |
Calculate the annual amount of interest earned on this account that will be deposited on January 1, 2015.
- Interest rates can fluctuate throughout the year. Working with the information from question 11, recalculate the annual amount of interest if all of the posted interest rates were adjusted as follows throughout the year:
Effective Date | Change | Percent |
---|---|---|
March 1 | Increase | 0.1% |
July 1 | Decrease | 0.15% |
September 1 | Increase | 0.25% |
November 1 | Decrease | 0.05% |
- Interest rates in the GIC markets are always fluctuating because of changes in the short-term financial markets. If you have $50,000 to invest today, you could place the money into a 180-day GIC at Canada Life earning a fixed rate of 0.4%, or you could take two consecutive 90-day GICs. The current posted fixed rate on 90-day GICs at Canada Life is 0.3%. Trends in the short-term financial markets suggest that within the next 90 days short-term GIC rates will be rising. What does the short-term 90-day rate need to be 90 days from now to arrive at the same maturity value as the 180-day GIC? Assume that the entire maturity value of the first 90-day GIC would be reinvested.
- A Citibank Savings Account posts the following simple interest rate structure. All interest is paid on entire balances, and interest is deposited to the account every quarter.
Balance | Interest Rate |
---|---|
$0–$2,500.00 | 0.05% |
$2,500.01–$5,000.00 | 0.1% |
$5,000.01 and up | 0.75% |
On January 1, 2013, the opening balance was $3,300. The following transactions took place during the first quarter (January 1 to March 31):
Deposits | Withdrawals |
---|---|
January 4 $350 | January 15 $500 |
January 29 $1,800 | January 19 $1,000 |
February 7 $2,300 | February 20 $1,300 |
March 2 $2,400 | March 7 $4,000 |
March 22 $1,100 | March 25 $2,000 |
Calculate the total amount of interest that will be deposited to the account on April 1, 2013.
- You are considering the following short-term GIC investment options for an amount of $90,000.
Option 1 | Option 2 | Option 3 | Option 4 | |
---|---|---|---|---|
Institution | Coast Capital Savings | MCAN Mortgage Corp. | DUCA Financial Services | Sun Life Financial Trust |
Term | 360 day | 2 × 180 day | 3 × 120 day | 4 × 90 day |
Posted Rate | 0.8% | 0.75% | 0.72% | 0.715% |
In all cases, assume that the posted rates remain unchanged and that the entire maturity value will be reinvested in the next short-term GIC. Calculate the total maturity value for each option at the end of 360 days.
8.4: Application - Promissory Notes
For each of the following questions, round all money to two decimals and percentages to four decimals.
Mechanics
- Examine this promissory note.
- Identify the six components of the promissory note.
- Determine the legal due date.
- Calculate the maturity value of the note.
For questions 2–12, solve for the unknown variables (identified with a ?) based on the information provided.
Issue Date | Term | Legal Due Date |
---|---|---|
2. January 16, 2011 | 320 days | ? |
3. November 29, 2010 | ? | November 1, 2011 |
4. ? | 120 days | April 11, 2012 |
Issue Date | Legal Due Date | Face Value | Term | Interest Rate | Maturity Value |
---|---|---|---|---|---|
5. October 4, 2011 | ? | $33,550 | 90 days | 7.1% | ? |
6. April 18, 2011 | ? | $72,950 | 150 days | ? | $75,014.09 |
7. January 29, 2011 | ? | ? | 270 days | 4.8% | $8,805.16 |
8. ? | December 23, 2011 | $29,900 | ? | 6.25% | $31,553.72 |
Issue Date | Face Value | Term | Interest Rate | Maturity Value | Discount Rate | Sale Date | Proceeds |
---|---|---|---|---|---|---|---|
9. March 2, 2011 | $4,500 | 180 days | 18.1% | ? | 21.5% | June 20, 2011 | ? |
10. Dec. 15, 2010 | ? | 350 days | 12.2% | $16,764.25 | 13.75% | ? | $16,196.81 |
11. July 3, 2011 | $43,000 | 165 days | 4.36% | ? | ? | Oct. 18, 2011 | $43,490.78 |
12. Nov. 12, 2010 | $18,750 | 340 days | ? | $19,904.10 | 8.35% | May 27, 2011 | ? |
Applications
- A $20,000 promissory note is issued on June 12 for a five-month term carrying an annual interest rate of 5.5%. Calculate the maturity value of the note.
- A noninterest-bearing note is issued on February 12 of a non–leap year in the amount of $14,250. It has a term of 10 months. It is sold on September 22 with a negotiated interest rate of 4%. Determine the proceeds of the sale.
- A $4,700 promissory note is issued on November 12 for an 11-month term (February is a leap year) carrying an annual interest rate of 12%. It is sold on July 3 with a negotiated interest rate of 16%. Determine the proceeds of the sale.
- A $12,600 promissory note was issued on March 13 with a term of 120 days. It acquired $541.37 of interest. What was the annual interest rate on the promissory note?
- A $34,250 promissory note with 5.9% annual interest issued on April 30 had a value of $35,296.36 on its legal due date.
- What is the term of the promissory note?
- If the note was sold to an investor on August 28 for $34,853.60, what was the negotiated rate of interest on the sale?
Challenge, Critical Thinking, & Other Applications
- An investor purchased a promissory note on June 15 for $27,329.96 using a negotiated interest rate of 10.5%. The promissory note had been issued on April 2 for a term of eight months at an interest rate of 9.25%. Calculate the original face value of the promissory note.
- A 150-day promissory note has 18.2% annual interest. Forty-five days after the issue, the owner of the note has two offers to purchase the note. The first offer is to purchase the note immediately at an annual interest rate of 19.3%. The second offer is to purchase the note 15 days later at an annual interest rate of 21%. If money can be invested at 5% simple interest, which offer should the owner accept? What percentage more proceeds are realized that way?
- Teagan is a good friend of the manager of Borling Industries and is looking to borrow $6,000 from the company on January 12, 2012, for a term of anywhere from 6 to 11 months. Both parties are agreeable to the following possibilities with respect to term and the corresponding interest rate.
Term | Interest Rate |
---|---|
6 months | 9% |
7 months | 7.75% |
8 months | 7% |
9 months | 6.15% |
10 months | 5.5% |
11 months | 5% |
- If Teagan wants to pay the least amount of total interest on his promissory note, which term should he pursue? What amount of interest will he pay?
- If Borling Industries wants to maximize the total interest on the promissory note, which term should it pursue? What amount of interest will it receive?
8.5: Application - Loans
Mechanics
Fill in the partial repayment schedules with the missing values.
- Woodgrain Industries took out an operating loan with RBC for $20,000 at a fixed interest rate of 8% on September 14. The operating loan requires a monthly fixed payment of $800 on the 14th of every month. Create the first three months of its repayment schedule.
Date | Balance before Transaction | Annual Interest Rate | Number of Days | Interest Charged | Accrued Interest | Payment (+) or Advance (−) | Principal Amount | Balance after Transaction |
---|---|---|---|---|---|---|---|---|
Sep 14 | $20,000 | |||||||
Oct 14 | 8% | $800 | ||||||
Nov 14 | 8% | $800 | ||||||
Dec 14 | 8% | $800 |
- Gayle has a HELOC with MCAP Financial Corporation at an interest rate of prime + 3%. Her current balance owing on November 1 is $13,750 and she is required to make interest-only payments on the first of every month. The prime rate is set at 3.75%. She makes one payment of $2,500 on January 19. Create three months of her repayment schedule.
Date | Balance before Transaction | Annual Interest Rate | Number of Days | Interest Charged | Accrued Interest | Payment (+) or Advance (−) | Principal Amount | Balance after Transaction |
---|---|---|---|---|---|---|---|---|
Nov 1 | $13,750 | |||||||
Dec 1 | 6.75% | |||||||
Jan 1 | 6.75% | |||||||
Jan 19 | 6.75% | $2,500 | ||||||
Feb 1 | 6.75% |
- Grant has a personal LOC with TD Canada Trust at prime + 4.5%, where the current prime rate is 3.75%. On March 31, his balance owing was $5,000. On April 4 he advanced $1,000 and on April 24 he paid $2,250. The prime rate declined by 0.5% on April 14. He is required to make the complete interest payment at the end of every month. Create the repayment schedule for April.
Date | Balance before Transaction | Annual Interest Rate | Number of Days | Interest Charged | Accrued Interest | Payment (+) or Advance (−) | Principal Amount | Balance after Transaction |
---|---|---|---|---|---|---|---|---|
Mar 31 | $5,000.00 | |||||||
Apr 4 | 8.25% | -$1,000 | ||||||
Apr 14 | 8.25% | |||||||
Apr 24 | 7.75% | $2,250 | ||||||
Apr 30 | 7.75% |
- Amarjeet graduated from the University of Calgary on May 2 and has student loans totalling $35,000. The prime rate upon graduation was 4.5%. He has decided to pay in full the interest charged during the grace period (i.e., he is not converting it to principal) before starting monthly payments of $600 at the fixed interest rate. Calculate the total interest paid during the grace period and the first three months of his repayment schedule.
Date | Balance before Transaction | Annual Interest Rate | Number of Days | Interest Charged | Accrued Interest | Payment (+) or Advance (−) | Principal Amount | Balance after Transaction |
---|---|---|---|---|---|---|---|---|
June 1 | $35,000 | |||||||
Nov 30 (inclusive) | 7% | $0.00 | ||||||
Dec 31 | 9.5% | $600.00 | ||||||
Jan 31 | 9.5% | $600.00 | ||||||
Feb 29 | 9.5% | $600.00 |
Applications
- A $7,500 demand loan was taken out on March 4 at a fixed interest rate of 7.72% with fixed monthly payments of $1,200. The first monthly repayment is due April 4 and the 4th of every month thereafter. Prepare a full repayment schedule for the loan.
- Paintball Paradise took out a $17,500 operating loan on February 15 (in a non–leap year) at an interest rate of prime + 3.5% when the prime rate was 3.75%. The loan requires fixed monthly payments of $3,000 each made on the 15th of the month starting with March 15. If Paintball Paradise plans to make an additional payment of $4,500 (without penalty) on May 4 and the prime rate rises by 0.75% on June 28, construct a full repayment schedule for the loan.
- Vertical Adventures has an open line of credit with a zero balance at its credit union using a fixed interest rate of 7.35%. On the last day of every month, the accrued interest must be paid. On July 8 and August 14, the company made advances of $15,000 and $12,000, respectively. On July 30, it made a payment of $10,000. Vertical Adventures will restore its zero balance on August 31. Construct a full repayment schedule from July 8 to August 31.
- Scotiabank approved a $250,000 line of credit for Buhler Industries at prime + 1%. It requires only the repayment of accrued interest on the 27th of each month, which is automatically deducted from the chequing account of Buhler Industries. Buhler took out an advance on December 2 for $200,000 and made a payment of $125,000 on January 12. The prime rate was 6.5% initially and increased to 7.5% on January 4. Calculate the total interest charged to Buhler Industries on December 27 and January 27.
- On May 9 Rainbow Daycare established an open line of credit with its bank and immediately withdrew $25,000. The interest rate was set at prime + 4.75%, and the prime rate was 6.75%. The line of credit requires a payment on the 9th of every month in the amount of "$1,000 or 5% of the current balance, whichever is greater." Rainbow Daycare took another advance of $15,000 on June 2. It made a payment of $28,000 on July 16. The prime rate decreased by 0.5% on June 30. Create a repayment schedule from May 9 to August 9 and calculate the total interest paid by Rainbow Daycare.
- Lacy has a $40,000 student loan when she graduates on May 4, and the prime rate is set at 5.25%. She has decided at the end of the grace period to convert the interest to principal, and she sets her fixed monthly payment at $850. She opts for the variable rate on her student loan. Create the first four repayments of her repayment schedule. Calculate the total interest charged for both the grace period and the four payments combined. Assume February does not involve a leap year.
- Genevieve has a $32,000 student loan when she graduates on September 20, at which time the prime rate is 6%. She has decided at the end of the grace period to convert the interest to principal, and she sets her fixed monthly payment at $910. She opts for the fixed rate on her student loan. The prime rate rises by 0.5% on November 3 and by another 0.5% on February 18. Create the first six repayments of her repayment schedule. Calculate the total interest charged for both the grace period and the six payments combined. Assume February has 29 days.
Challenge, Critical Thinking, & Other Applications
- Home Run Sports has an operating loan with CIBC. The interest rate is set at prime + 3.25%, and on the 17th of every month a payment in the amount of 4% of the current balance is automatically taken from the borrower’s chequing account. The prime rate is currently 5.75%. On March 7, Home Run Sports took a first advance of $6,000. It took additional advances of $10,000, $8,000, and $12,000 on March 28, April 17, and May 24, respectively. It made payments of $4,000, $2,500, and $10,000 on April 10, May 17, and May 29. The prime rate increased by 0.25% on March 31 and by an additional 0.25% on May 31. Create a repayment schedule and calculate the total interest paid from March 7 to June 17.
- Eyeland Optical has a line of credit with its credit union with an established credit limit of $24,000 and interest set at prime + 4.1%. Eyeland Optical is allowed to exceed its credit limit, but if it does so the entire line of credit becomes subject to 28% interest. Payment terms require the repayment of the accrued interest only on the 11th of every month. On July 8, Eyeland Optical made its first withdrawal of $10,000. It made further advances of $15,000, $15,000, and $8,000 on August 11, August 20, and August 28, respectively. It made payments of $3,500, $2,000, and $28,000 on July 31, August 15, and September 11, respectively. On October 11, Eyeland Optical restored a zero balance. The prime rate was initially 3.5%, rose by 0.25% on July 30, and declined by 0.5% on September 1. Create a repayment schedule and calculate the total interest charges from July 8 to October 11.
- Pierre has a $40,000 student loan when he graduates on December 22. The prime rate is currently 4.25%. He decides that at the end of the grace period he will convert the interest to principal, take the variable interest rate, and set his fixed monthly payment at $750. The prime rate rises by 0.5% on each of February 15, April 15, June 30, August 15, and November 30. He makes additional payments of $500 on each of September 3, October 21, and December 6. Create a repayment schedule and calculate the total interest charged to Pierre from January 1 to December 31. Assume February has 29 days.
- Use the following information for a student loan (assume February does not involve a leap year):
Prime Rate: 5% on June 1, rising by 0.25% on September 13, rising by 0.5% on January 25, rising by 1% on March 18.
Grace Period: June 1 to November 30 (inclusive)
Total Student Loan as of June 1: $13,000
Chosen Fixed Payment: $350
There are essentially four options on this loan:
Option | Grace Period Interest on November 30 | Interest Rate |
---|---|---|
1 | Pay it off | Variable |
2 | Convert to principal | Variable |
3 | Pay it off | Fixed |
4 | Convert to principal | Fixed |
For each option, create a complete repayment schedule and calculate the total interest charges. How much less interest than the worst option does the best option incur?
8.6: Application - Treasury Bills & Commercial Papers
For each of the following questions, round all money to two decimals and percentages to four decimals.
Mechanics
For questions 1–7, solve each of the following T-bills or commercial papers for the unknown variables (identified with a ?) based on the information provided.
Purchase Price on Date of Issue | Term (days) | Yield | Face Value |
---|---|---|---|
1. ? | 90 | 1.75% | $100,000 |
2. $240,663.83 | 364 | ? | $250,000 |
3. ? | 270 | 5.67% | $500,000 |
4. $295,796.45 | 182 | ? | $300,000 |
5. $73,307.58 | ? | 4.63% | $75,000 |
6. $9,931.92 | ? | 2.78% | $10,000 |
7. $190,175.88 | 364 | 5.18% | ? |
Applications
- A 60-day, $90,000 face value commercial paper was issued when yields were 2.09%. What was its purchase price?
- A Government of Canada V39065 issue 90-day T-bill achieved its highest rate of return on May 24, 2000, with a yield of 5.74%. It realized its lowest rate of return on February 26, 2010, with a yield of 0.16%. Calculate the purchase price of a $100,000 T-bill on each of these dates. In dollars, how much more yield did an investor realize in 2000 than in 2010?
- A Government of Canada V121780 issue 364-day T-bill achieved its highest rate of return on May 17, 2000, with a yield of 6.31%. It realized its lowest rate of return on May 6, 2009, with a yield of 0.42%. Calculate the purchase price of a $55,000 T-bill on each of these dates. In dollars, how much more yield did an investor realize in 2000 than in 2009?
- Pawan is the marketing manager for Cyanamid Canada. His company will execute a marketing program half a year from now that requires a $500,000 investment. If his finance department had $485,000 to invest today into a 182-day $500,000 face value commercial paper yielding 4.55%, would it have enough money to purchase the commercial paper? Show calculations to support your answer.
- A 182-day Province of Manitoba T-bill with a face value of $250,000 was issued 102 days ago, when the yield was 4.3%. What is its purchase price today if the current rate of return is 4.53%?
- William purchased a $110,000 270-day commercial paper on its date of issue, when the yield was 3.89%. He sold it 178 days later when yields had increased to 4.03%. How much money did William earn on his investment?
- Dollar Thrifty Automotive Group issued a $1,000,000, 180-day commercial paper. A bank purchased the paper for $975,560.21 on the issue date. What was the yield for the commercial paper on its issue date?
- A 90-day Province of Ontario T-bill with a $35,000 face value matures on December 11. Farrah works for Hearthplace Industries and notices that the company temporarily has some extra cash available. If she invests the money on October 28, when the yield is 4.94%, and sells the T-bill on November 25, when the yield is 4.83%, calculate how much money Farrah earned and the rate of return she realized.
- On August 8, Harriet invested in a $150,000 Canadian Wheat Board commercial paper on its date of issue with a 220-day term at a yield of 5.98%. On October 15, she sold the commercial paper to another investor when the current market rate was 5.75%. Calculate the amount of money earned and the rate of return realized.
Challenge, Critical Thinking, & Other Applications
- Philippe purchased a $100,000 Citicorp Financial 220-day commercial paper for $96,453.93. He sold it 110 days later to Damien for $98,414.58, who then held onto the commercial paper until its maturity date.
- What is Philippe’s actual rate of return?
- What is Damien’s actual rate of return?
- What is the rate of return Philippe would have realized if he had held onto the note instead of selling it to Damien?
- Comment on your answers to the above.
- It is interesting to note another application of the percent change Formula 3.1. Recall, \(\Delta \%=\dfrac{\text { New-Old }}{\text {Old }} \times 100\). In the case of simple interest and solving for rate of return, assign \(S = \text{New}\), \(P = \text{Old}\), and \(r = Δ\%\). Therefore, you rewrite the formula as \(r=\dfrac{S-P}{P} \times 100\). It is critical to recognize that the \(r\) is based on the implied time period between the \(S\) and \(P\). If the \(S\) is 100 days after the \(P\), then the \(r\) is a “% per 100 days.” Since \(r\) is expressed annually, you must convert it by dividing by the period and multiplying by 365. Recalculate questions 2 and 4, showing how to arrive at the same answer by using the percent change formula.
Spreadsheet Applications
- Calculate the purchase price of a $10,000 T-bill with a yield of 2.95% using different terms of 30 days, 60 days, 90 days, 182 days, and 364 days. What do you observe about the purchase price when comparing these various terms?
- Take a $100,000, 364-day T-bill that had a rate of return of 4.73% upon issue.
- Calculate the purchase price on the date of issue, and also calculate the purchase price if the rate of return had been higher or lower by 1% on the date of issue.
- Repeat part (a) on the 120th day and the 240th day after the date of issue.
- Compare your answers to (a) and (b) above and comment on your findings.
Review Exercises
Mechanics
- If $4,000 is borrowed from April 3 to June 22 at a simple interest rate of 3.8%, how much interest is paid on the loan?
- A savings account pays flat-rate interest of 1.45%. If a balance of $3,285.40 is maintained for the entire month of August, how much interest does the savings account earn?
- What is the legal due date and maturity value on that date for a 125-day $51,000 promissory note issued on May 14 at
- A full-time student graduated from college on December 16, 2013, with $26,500 in outstanding student loans. Calculate the interest accrued during his grace period if the prime rate is 4.7%.
- A 182-day $1,000,000 Government of British Columbia T-bill was issued when the market rate of return was 4.21%. Calculate the purchase price of the T-bill on its issue date.
- If you place $8,000 into a 300-day short-term GIC at Scotiabank earning 0.95% simple interest, how much will you receive when the investment matures?
Applications
- Proper accounting procedures require accountants to separate principal and interest components on any loan. Allocate the principal and interest portions of a $24,159.18 payment clearing a 147-day loan at 8.88%.
- Cadillac Fairview withdrew $115,000 from its operating loan account on September 4 to perform some needed maintenance on one of its properties. The operating loan requires interest at prime + 3% and fixed $25,000 monthly payments starting October 1. The company thinks it can make an additional payment of $35,000 on November 18. The prime rate on the date of withdrawal was 3.6%, and it increased by 0.5% on October 27. Construct a full repayment schedule for this loan.
- As part of your financial plan for retirement, you purchased a 270-day $25,000 commercial paper on its date of issue, July 14, when market yields were 2.94%. 234 days later, you sold the note when market yields were 2.76%. What rate of return did you realize on your investment?
- Alterna Savings and Bank offers a Daily Interest Savings Account with posted interest rates as indicated in the table below. The entire balance qualifies for the posted rate, and interest is calculated daily based on the closing balance.
Balance | Interest Rate |
---|---|
$0–$5,000.00 | 0.10% |
$5,000.01–$10,000.00 | 0.15% |
$10,000.01–$25,000.00 | 0.20% |
$25,000.01 and up | 0.25% |
BOS Designer Candles Inc. has an opening balance in July of $17,500. Three deposits in the amounts of $6,000, $4,000, and $1,500 were made on July 4, July 18, and July 22, respectively. Two withdrawals in the amount of $20,000 and $8,000 were made on July 20 and July 27, respectively. What interest for the month of July will Alterna deposit to the account on August 1?
- Shannon has a $68,000 student loan when she graduates on August 28, 2014, and the prime rate is set at 4.9%. She will convert the interest to principal at the end of her grace period, and she elects to take the variable rate on her student loan. She sets her fixed monthly payment at $1,400. The prime rate decreases by 0.5% on January 15 and rises by 0.75% on April 25. Compute the first six repayments of her repayment schedule. Calculate the total interest charged for both the grace period and the six payments combined.
- Sturm put $48,700 into a 10-month term deposit, but needed to withdraw the funds after five months to deal with a family emergency. The credit union penalized him 2.35% off of his interest rate for the early withdrawal and deposited $49,602.98 into his account.
- What was Sturm's original interest rate?
- How much interest, in dollars, was he penalized for the early withdrawal?
- Jerry's Concrete allows his customers to create six-month promissory notes on any stamped concrete driveway project with interest at 8.9%. Three months after completing a $17,300 job on March 2, 2014, Jerry had some liquidity problems and sold the promissory note to a financial institution at an interest rate of 10.35%. Calculate Jerry's proceeds on the sale.
- Three hundred days from now you will be departing on a backpacking trip through Europe. You need $4,000 in spending money to take with you. Today, you currently have saved $3,960.
- If you place your money in consecutive 100-day short-term GICs earning 1.02%, will you meet your goal? Assume the full maturity values are reinvested into the next GIC.
- What would the interest rate need to be if you had placed your money into a single 300-day short-term GIC to reach your goal?
Challenge, Critical Thinking, & Other Applications
- A credit union posts the following tiered interest rate structure for its savings accounts. Only each tier is subject to the posted rate and is computed using the daily opening balance in the account. Interest is deposited to the account on the last day of every month.
Balance | Interest Rate |
---|---|
$0–$4,000.00 | 0.15% |
$4,000.01–$8,000.00 | 0.25% |
$8,000.01 and up | 0.45% |
On August 1, the opening balance was $6,400. Three deposits of $2,000, $3,500, and $1,500 were made on August 3, August 10, and August 27, respectively. Two withdrawals of $7,000 and $1,900 were made on August 6 and August 21, respectively. Compute the total interest earned for the month of August.
- Island Lakes Dental (ILD) found itself with an urgent need to replace its X-ray machine when its existing machine suddenly became inoperable. It borrowed $43,000 on May 12 from its operating loan with interest set at prime + 2.4%. The prime rate was 2.75% and increased by 0.5% on June 5. The loan requires payment of the balance in full when the balance is less than $5,000, or 10% of the current balance on the first of every month. ILD made payments of $15,000 on May 29 and $21,000 on June 21. Create a full repayment schedule for its operating loan and calculate the total interest charged on the purchase.
- Pendragon Inc. has an operating loan with a balance of $52,000 on September 1 with interest set at prime + 4.75%. On the first of every month the operating loan requires repayment of the accrued interest only. The current prime rate is 4.75% and will decrease to 4.25% on October 7. On September 15, it has a 270-day $20,000 GIC with 3.8% interest maturing. On October 15, it has a 320-day $18,000 GIC with 3.68% interest maturing. It will use these maturing investments to pay down its operating loan. Construct a repayment schedule from September 1 to November 1 only.
- A 364-day, $50,000 face value T-bill is issued when market yields are 2.85%. The T-bill is sold to another investor every 91 days until maturity, with yields of 3.1%, 2.98%, and 3.15% on each of the dates of sale, respectively. Compute the purchase price for each investor, including the date of issue. For each investor, calculate the actual rate of return realized on their investment.
- Brant and Sylvia have a prime + 2.35% HELOC with a $30,000 balance on March 1 with accrued interest payable on the first of every month. They also have a savings account with a $10,000 balance on March 1 earning 0.85% payable on the first of every month. On March 18 they deposited $4,000 into the savings account. On both March 5 and March 30 they transferred $3,000 from their savings to their HELOC. Prime was initially at 3.75% but increased to 4.25% on March 22. Calculate the balance on April 1 in both the savings account and the HELOC.
- Norbert is looking to make a short-term investment for 120 days with the $25,000 he just inherited from his father’s estate. His options are as follows. Compute the maturity values of each of the options and rank his choices.
- Two back-to-back 60-day GICs earning 2.25% today, with a forecasted rate of 2.5% 60 days from now. Assume the full maturity value is rolled over into the next investment.
- One 120-day GIC earning 2.35%.
- A 90-day $25,000 face value T-bill earning 2.28% followed by a 30-day $25,000 face value T-bill earning a projected 2.38%. Assume any leftover funds are invested to earn 0.85% interest.