Review Questions for Chapter 4

Click on the question number to get to the solution.

^[1] Complete the following table, assuming an interest rate of 10% compounded quarterly.

^[2] Make a table with the same headings as previous, but for a loan of $12,000 for 5 months at 15% compounded monthly.

^[3] Write the following nominal rates in the “j_m” notation and find the corresponding periodic rates, i (include the period with these). The first line is completed as an example.

^[4] Write each of the following rates as nominal rates and complete the table.

^[5] Complete the following table, using the compound-interest formula to calculate the future value of each loan.

^[6] Government compound-interest savings bonds have the interest compounded every year. Suppose that a $1,000 bond paid interest at 8.5% compounded annually and was kept for three years. Find the value (Future Value) of the bond at the end of the three years by:

1. Showing the interest and balance each year.
2. Using the compound-interest formula to get FV at the end of two years.

^[7] A loan of $24,000 for two years is to carry interest at 14% compounded semi-annually. Use the compound-interest formula to find the value of the loan at the end of two years.

^[8] A “junk” bond was supposed to pay interest at 25% paid annually. Unfortunately, no payments were made for the last seven years, so the interest was allowed to compound at 25% compounded annually.

1. Find the value at the end of each of the seven years for a bond with a principal of $1,000 at the start.
2. Also find the value the loan would have had at the end of each year if the interest had been simple interest at 25% annually. Note the difference caused by compounding.

^[9] Find the present values of each of the amounts below, filling in the rest of the table when you do so.

Check your last result by assuming the PV was invested for one year in an account paying 10% compounded quarterly and adding on the interest each period as in Problem 1.

^[10] According to the terms of his uncle’s will, Tom Jones is to receive $50,000 2.5 years from now. Tom would like to borrow as much as he can now and pay it off with his inheritance. Use the compound-interest formula to find out how much he can borrow if the interest rate is as follows:

1. 10% compounded monthly.
2. 9% compounded monthly.
3. 8% compounded monthly.

^[11] Ann Lee has a lease on a government property which must be paid by a lump sum of $6,000 every year. The next payment is due in 10 months from now, and Ann plans to invest enough money in an account at her bank so that the amount in the account in 10 months will cover the $6,000 payment. If the interest rate is 8.5% compounded monthly, how much should she place in the account?

^[12] Ajax Company is borrowing $100,000 now and has agreed to pay 11% compounded annually on the outstanding balance at all times. Ajax plans to pay $30,000 at the end of the first year of the loan and $35,000 at the end of the second year. The remaining debt is to be completely paid off by a single payment at the end of the third year. Draw a cash-flow diagram and find the amount that should be paid at the end of the third year by:

1. Finding the interest and balance due at the end of each year.
2. Using an equation of value at a focal date (e.g., at year 3).

^[13] AC Holdings has taken over a company with a debt that was to be paid by a payment of $85,000 one year from now. Instead, AC has agreed with the holder of the debt to pay it off early and be allowed 12% compounded quarterly for early payment. It plans to pay $40,000 now and the rest in six months.

1. Draw a cash-flow diagram and find the amount that should be paid in 6 months.
2. Repeat (a) but assume the payments now and in 6 months are to be equal in size.

^[14] North Credit Union advertises that it will pay 7% compounded annually on money on deposit for periods longer than one year and that the money may be taken out, with interest, at any time after the first year. A depositor placed $20,000 on deposit at the above rate for four years, but decided to withdraw it after 2.5 years.

1. How much should the depositor receive?
2. Show that, if this amount were deposited at the same rate for the remaining 1.5 years, the result would be the same as if it had never been withdrawn.

^[15] A loan of $17,000 for three years resulted in a future value of $21,560.11. Use your calculator functions to find the nominal interest rate if the loan was compounded:

1. 1. quarterly
   2. semi-annually.

^[16] Use your calculator functions to find the FV of a principal of $1.00 invested for one year at 16% compounded quarterly, and leave the results in your calculator. Then find the nominal monthly rate that will give the same FV – the rate compounded monthly that is equivalent to 16% compounded quarterly. Check your answer by finding the future value of a principal of $10,000 invested for two years at each rate. You should get the same answer in each case.

^[17]     Complete  the following table of equivalent nominal rates, giving answers in percent to six decimals.  Each row will contain equivalent rates.

^[18]  Use your results in Row 1 in the previous problem to find the future value of a loan of $4,000 for 18 months by doing the compounding:

1. monthly.
2. Quarterly.
3. Semi-annually
4. Annually

^[19]  Your first child was born this year and you decide to save for her education. You deposit $2,000 into an RESP (registered education savings plan) that pays 4.5% compounded annually (j₁=0.045). How much will your child have in 18 years? Use both the formula and the TVM Keys.

^[20] You borrow $5,000 from a private loan company and agree to pay it back with interest calculated at 10% compounded quarterly. How much will you owe in 30 months? Use both the formula and the TVM Keys.

^[21] You are expecting a tax refund of $3,000 six months from now. You take your T4 slips to H&P Square Tax preparation service and they agree to give you the money now if you sign over the refund to them. How much money will you receive today if interest is calculated at j₁₂ = 9$%. Use both the formula and the TVM Keys.

^[22] You go to purchase a brand new motorcycle and the dealer quotes you a price of $14,000 to be paid as a single payment in 30 months. You would like to pay cash today for the motorcycle. How much should you offer if interest is calculated at 8% compounded semi­ annually? Use the formula and the TVM Keys.

^[23] You have $75,000 and decide to purchase a 30 year Government of Canada strip bond with an interest rate of 7.0% compounded semi-annually. How much will the bond be worth in 30 years? (i.e., what is the maturity value of the strip bond?)

^[24] Financial planners predict you will need one million dollars to retire.

1. You would like to buy a 30-year strip bond with an interest rate of 7% compounded semi-annually that has a maturity value of one million dollars. How much will you need to pay for this strip bond today?
2. Unfortunately, you only have $75,000 available to buy a bond. You are considering buying a junk bond (risky bond) because you know that the interest rates are much. higher than for a Government of Canada bond. What nominal interest rate, compounded semi-annually, do you require so that the bond will have a maturity value of $1,000,000 in 30 years?
3. Your friend convinces you that junk bonds are too risky to be used as a retirement investment. If instead, you buy the Government of Canada bond (with your $75,000) that pays 7% compounded semi-annually, how many years will it take you to reach your goal of having $1,000,000?

^[25]  If an investment grows from $10,000 to $16,000 in 27 months, what was the nominal rate of interest, compounded quarterly?

^[26]     If an investment grows from $4,000 to $6,000 in 48 months, what was the nominal rate of interest:

1. compounded monthly?
2. compounded quarterly?
3. compounded semi-annually?
4. compounded annually?

^[27] You would like to save to return to school. You deposit $4,000 into a GIC that pays . You have decided to return to school when your savings grows to at least $6,000. If you make no more contributions, how many years will it take you to reach your goal?

^[28] How many years will it take $300.00 to accumulate to $425.29 at 7% compounded monthly?

^[29] An investment of $1,500.00 made 27 months ago is now worth $1753.48. What nominal rate of interest, compounded quarterly, did this investment earn?

^[30] You have always wanted to go to Orlando, Florida. You estimate you will need $6,000 for your trip. You deposit your tax refund of $5,016.10 into an account that pays 12% compounded monthly. How many years will it take to reach $6,000?

^[31] You put $5,000 into a 3-year term deposit that pays interest at 6.3% compounded quarterly. After the end of the three years you renew the term deposit plus accumulated interest at 7.2% compounded semi­ annually for an additional three years.

1. How much money will you have at the end of the six years?
2. How much interest did you earn?

^[32] You borrowed some money from the Honest Shark Finance Co. You made only one payment of $25,292.49 at the end of 5 years to pay off the loan. The interest rate was 18.5% compounded semi­ annually for the first 2 years and 19.6% compounded quarterly for the remaining years.

1. How much did you borrow?
2. How much interest did you pay?

^[33] You are going to purchase a car and are given two options to pay.

• You can pay $18,000 in cash today, or
• $5,000 after one year, and a second payment of $15,000 in 2 years (from today).

Which option is better? Your answer  should  be stated in terms of today’s dollars. The prevailing  interest  rate is 7% compounded monthly (j₁₂ = 7%)?

^[34]    You have a line of credit that charges interest at j₁₂ = 8%. You borrowed $4,000 six months ago and $2,000 two months ago. You would like to repay the loan with a single payment in 6 months time. Calculate the size of the payment. Use 6 months as the focal date.

^[35] Repeat the above question with you making two equal payments at six and twelve months. Find the size of the equal payments. Use 6 months as the focal date. Note: To save time use some of the values you calculated above since the focal date is the same.

^[36] You purchased a machine for your plant and the contract calls for equal payments of $12,000 in 12 months and 24 months. Your cash flow is better than you projected so you would like to repay the loan early with a single payment today. Calculate the size of the payment if interest is 9% compounded semi­ annually. Use today as the focal date.

^[37] Repeat the above question with you making two equal payments, one today and one in six months time. Find the size of the equal payments. Use today as the focal date. Note: To save time use some of the values you calculated above since the focal date is the same.

^[38] You were supposed to pay $5,000 today. Up until yesterday you had the money but you lost it all gambling at the Great American Casino. You arrange with the bank to defer the payment. You will pay $2,000 at the end of 18 months and 3 equal payments at the end of 24 months, 30 months and 36 months . The interest rate is 10% compounded quarterly (j₄ = 10%).

1. Find the size of the equal payments. Use 30 months as the focal date.
2. How much interest did you pay as a result of gambling and losing the $5,000?

^[39] You borrowed $12,000 2.5 years ago. You agreed to repay the loan with one payment 15 months from today, and a second payment, $3,000 larger than the first, 27 months from today. Find the size of each payment if money is worth 11% compounded quarterly? Use 27 months as the focal date.

^[40] You have $10,000 to invest today. How much would you have (to the nearest $100) at the end of 30 years if your money earns:

1. 12% interest, compounded monthly?
2. 12% simple interest?

^[41] Your Credit Card charges you a nominal interest rate of 18.6% per year. If they compound interest monthly, what effective rate of interest do they charge?

^[42]     You are offered two options for your mortgage.

• A Canadian bank offers you a rate of j₂ = 8.40%
• A US bank offers you j₁₂ = 8.30%.

Which rate is better? Convert both rates to effective rates to compare them.

^[43] What is the effective rate of interest earned on an investment of $10,652,952,497,853.65 that earns 15% compounded quarterly?

^[44] What nominal rate, compounded semi-annually, is equivalent to 8% compounded quarterly?

^[45] What nominal rate, compounded quarterly, is equivalent to 8% compounded monthly?

^[46] What is the effective interest rate of 8% compounded monthly?

^[47] Canada Premium Bonds are a new kind of Canada Savings Bond. They were available for sale until November 1, 2022. They proposed to pay the following interest rates, compounded annually.

What effective interest rate would a Canada Premium Bond purchaser average over the five years? Hint: use the formula to find the FV. Do not round the FV.

^[48] What nominal rate, compounded monthly, is equivalent to 10% compounded semi-annually?

^[49] What is the effective interest rate of 20% compounded quarterly?

^[50] You have $5,000 saved. You are considering two investments:

• Canada Savings Bonds (CSB) which pay 5.25% in the first year, 6% in the second year, and 6.75% in the third year, compounded semi-annually.
• A 3-year “Bond-Beater” Guaranteed Investment Certificate (GIC) offered by a bank that pays 5.75%, 6.5%, and 7.25% compounded annually in the three successive years.

1. How much would you have at the end of 3 years if· you bought a $5,000 CSB and how much if you bought a $5,000 GIC?
2. Find the average effective interest rate earned for the entire 3-year period for both the CSB and the GIC.

^[51]     You invest $6,000 in a mutual fund. Your investment of $6,000 earns the following returns.

1. How much would your $6,000 investment be worth at the end of the fourth year?
2. How much did you earn (in dollars) during those four years?
3. In the fifth year the mutual fund loses money and the value of your investment decreases by $400. Calculate the average annual rate of return, compounded semi-annually, for the five years you have held your investment.
4. A friend has invested in a different mutual fund and says she doubled her money in seven years. What effective interest rate did she average over the seven years?

^[52]     A Canada  Savings  Bond  pays j₂ = 5% in the first year, j₂ = 8% in the second year, and j₂ = 10% in the third year. What nominal interest rate, compounded quarterly, would  provide the same return?  Do not round the FV.

^[53] An investment earned 12%, compounded quarterly, for two years and 10% compounded annually for the next three years. Calculate the average annual rate of return, compounded monthly, for the five years. Do not round the FV.

^[54] An investment earned 20%, 15%, -10%, 25%, and -5% in 5 successive years. What average annual rate of return, compounded annually, was earned for the entire 5-year period? Hint: use the formula to find the FV. Do not round the FV.