25. Thomas leases a car worth $24,000 at 2.99% compounded monthly. He agrees to make 36 lease payments of $330 each at the beginning of every month. What is the buyout price (residual value) of the car at the end of the lease? 26. Valentina leases a car worth $28,000 at 4.75% compounded monthly. She agrees to make 48 lease payments of $450
each at the beginning of every month. What is the buyout price (residual value) of the car at the end of the lease? 27. Ethan converted his RRSP into a RRIF that pays him $1500 at the beginning of every month for three years
followed by $2000 at the beginning of every month for the next five years. If the rate of return of the annuity is 3.85% compounded monthly during the entire period, what was the value of the RRIF?
28. Samuel is considering purchasing an annuity with a rate of return of 6% compounded quarterly which pays 20 equal payments of $1000 at the beginning of every 3 months, followed by another 20 equal payments of $1500 at the beginning of every 3 months. How much should he pay for the annuity?
In a general annuity due, payments are made at the beginning of each payment period, and the compounding period is not equal to the payment period. In this section, you will learn how to calculate the future value and present value of a general annuity due.
Similar to the ordinary general annuities, we first need to calculate the equivalent periodic interest rate per payment period (i2) using Formulas 10.3(a) and 10.3(b):
c =
Number of payments per year Number of compounding periods per year
i2 = (1 + i)c - 1
Then, substituting the value of 'i2' for 'i' in the simple annuity due Formula 10.4(a) and Formula 10.4(b), solve for 'FVDue' and 'PVDue', as follows:
FVDue = PMT i i 1 n 1 2 2 + -^ h
; E (1+ i2) and PVDue = PMT 1 1i i n 2 2 -^ + h -; E (1+ i2) Note: In using the Texas Instruments BA II Plus calculator for annuity due calculations, set the calculator to beginning of period calculations ('BGN' mode) as described in Section 10.2.
Calculating the Future Value, Amount Invested, and Total Interest of a General Annuity Due
Tao invested $5000 in a fund at the beginning of every three months for five years. The fund was earning an interest rate of 5% compounded monthly. (i) What was the total amount invested? (ii) What was the accumulated value of the investment? (iii) What was the interest earned over the period? This is a general annuity due as:
■
Payments are made at the beginning of each payment period (quarter)■
Compounding period (monthly) ≠ payment period (quarter)10.5
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Future Value and Present Value of a
General Annuity Due
Example 10.5(a)
(i) The total amount invested in the fund = n(PMT) = 20 # 5000.00 = $100,000.00 (ii) The accumulated value of the investment
c =
Number of payments per year
Number of compounding periods per year =
4 12 i2 = (1 + i)c - 1 = (1 + 0.004166...)(12/4) - 1 = 0.012552... per quarter Using Formula 10.4(a) and substituting i2 for i, FVDue = PMT i i 1 n 1 2 2 + -^ h ; E (1+ i2) = 5000 1 0.012552... 1 0.012552... 20 + -^ h ; E(1+ 0.012552) = 5000 [22.574503...] (1.012552...) = 114,289.3094... Therefore, the accumulated value of his investment would be $114,289.31. (iii) The interest earned over the period = FVDue - n(PMT)
= 114,289.31 - 100,000 = $14,289.31
Therefore, the interest earned over the period is $14,289.31.
Calculating the Present Value of a General Annuity Due
Alexandria inherited money that was invested in an account which provided her $4500 at the beginning of every month for 30 years. If the interest rate on the savings account was 4% compounded semi-annually, what was the amount of the inheritance? Round your answer to the nearest hundred dollars. This is a general annuity due as:
■
Payments are made at the beginning of each payment period (monthly)■
Compounding period (semi-annually) ≠ payment period (monthly) c =Number of payments per year Number of compounding periods per year
= 12 2 Solution continued Example 10.5(b) Solution
(i) The total amount invested in the fund = n(PMT) = 20 # 5000.00 = $100,000.00 (ii) The accumulated value of the investment
c =
Number of payments per year
Number of compounding periods per year =
4 12 i2 = (1 + i)c - 1 = (1 + 0.004166...)(12/4) - 1 = 0.012552... per quarter Using Formula 10.4(a) and substituting i2 for i, FVDue = PMT i i 1 n 1 2 2 + -^ h ; E (1+ i2) = 5000 1 0.012552... 1 0.012552... 20 + -^ h ; E(1+ 0.012552) = 5000 [22.574503...] (1.012552...) = 114,289.3094... Therefore, the accumulated value of his investment would be $114,289.31. (iii) The interest earned over the period = FVDue - n(PMT)
= 114,289.31 - 100,000 = $14,289.31
Therefore, the interest earned over the period is $14,289.31.
Calculating the Present Value of a General Annuity Due
Alexandria inherited money that was invested in an account which provided her $4500 at the beginning of every month for 30 years. If the interest rate on the savings account was 4% compounded semi-annually, what was the amount of the inheritance? Round your answer to the nearest hundred dollars. This is a general annuity due as:
■
Payments are made at the beginning of each payment period (monthly)■
Compounding period (semi-annually) ≠ payment period (monthly) c =Number of payments per year Number of compounding periods per year
= 12 2 Solution continued Example 10.5(b) Solution i2 = (1 + i)c - 1 = (1 + 0.02)(2/12) - 1 = 0.003305... per month Using Formula 10.4(b) and substituting i2 for i, PVDue = PMT 1 1i i n 2 2 -^ + h -; E (1+ i2) = 4500 -. . 0 003305 1 1 0 003305 360 f f -^ + h ; E (1 + 0.003305...) = 4500 [210.296672...](1.003305...) = $949,463.504172... Therefore, she inherited $949,500.00 (rounded to the nearest $100).
Choosing Between Lease and Buy Options and Calculating the Book Value
A supplier of machinery provided a company with the following two options to lease or buy equipment:
■
Buy Option: Pay $17,500 immediately to own the equipment.■
Lease Option: Make lease payments of $180 at the beginning of every month for four years. At the end of four years, pay $10,000 to own the equipment. The cost of borrowing is 3% compounded annually. (i) Which option is economically better for the company? (ii) In the lease option, what will be the book value of the equipment at the end of three years? The lease payments form a general annuity due as:■
Payments are made at the beginning of each payment period (monthly)■
Compounding period (annually) ≠ payment period (monthly) PVLease option = PVPMT + PVResidual Value (i) Evaluating Lease or Buy Options c =Number of payments per year Number of compounding periods per year
= 12 1 i2 = (1 + 0.03)(1/12) - 1 = 0.002466... per month Substituting 'i2' in Formula 10.2(b) PVDue = PMT 1 1i i n 2 2 -^ + h -; E (1+ i2) PVPMT = 180 0.002466... 1 - 1] +0.002466...g-48 < F(1+0.002466...) = 180[45.215228...](1.002466...) = 8158.813519... Solution continued Example 10.5(c) Book value of an asset (that is leased) is its value on a specific date and can be calculated by finding the present value of the remaining lease payments at that time.
Using Formula 9.1(b), PV = FV(1 + i)-n PV$10,000 = 10,000(1 + 0.03) -(1#4) = 10,000(0.888487...) = 8884.870479... PVLease Option = PVPMT + PV$10,000 = 8158.813519...+ 8884,870479... = $17,043.684... The present value of the lease option is less than the purchase price of $17,500. Therefore, the lease option is economically better. (ii) Calculating the book value at the end of three years n = Remaining lease payments = 12 monthly payments Book Value Three years = PVRemaining payments + PV$10,000
= 180 <1 - 1]0.002466...+0.002466...g-12F(1 + 0.002466...) + 10,000(1 + 0.03)-(1 × 1) = 2131.011158 + 9708.737864 = 11,839.74902 = $11,839.75 Therefore, in the lease option, the book value of the equipment at the end of three years will be $11,839.75.
Using the financial calculator, you can either compute
‘PV’ separately and add the
answers together, or you can do the entire calculation in one step as shown here.
1. Calculate the future value of the following general annuities due:
Periodic Payment Payment Period Term of Annuity Interest Rate Compounding Frequency
a. $3750 Every year 16 years 4.80% Monthly
b. $1950 Every 6 months 12.5 years 4.00% Quarterly
c. $1500 Every 3 months 9 years and 9 months 3.75% Daily
d. $1200 Every month 6 years 7 months 3.90% Semi-annually
2. Calculate the future value of the following general annuities due:
Periodic Payment Payment Period Term of Annuity Interest Rate Compounding Frequency
a. $3600 Every year 6 years 4.50% Daily
b. $1250 Every 6 months 9.5 years 5.10% Quarterly
c. $2750 Every 3 months 5 years and 9 months 4.60% Monthly
d. $1950 Every month 12 years 8 months 6.00% Semi-annually
3. Calculate the present value of each of the general annuities due in Problem 1. 4. Calculate the present value of each of the general annuities due in Problem 2.
Using Formula 9.1(b), PV = FV(1 + i)-n PV$10,000 = 10,000(1 + 0.03) -(1#4) = 10,000(0.888487...) = 8884.870479... PVLease Option = PVPMT + PV$10,000 = 8158.813519...+ 8884,870479... = $17,043.684... The present value of the lease option is less than the purchase price of $17,500. Therefore, the lease option is economically better. (ii) Calculating the book value at the end of three years n = Remaining lease payments = 12 monthly payments Book Value Three years = PVRemaining payments + PV$10,000
= 180 <1 - 1]0.002466...+0.002466...g-12F(1 + 0.002466...) + 10,000(1 + 0.03)-(1 × 1) = 2131.011158 + 9708.737864 = 11,839.74902 = $11,839.75 Therefore, in the lease option, the book value of the equipment at the end of three years will be $11,839.75.
Using the financial calculator, you can either compute
‘PV’ separately and add the
answers together, or you can do the entire calculation in one step as shown here.
for 2 years and 9 months and the fund was earning 2.85% compounded monthly?
6. As part of his New Year's resolution, Clive committed to save $500 per month in a mutual fund account. He deposited the money at the beginning of every month and projected that he would earn 8.5% compounded annually on the funds. At the end of 12 months, how much did Clive accumulate? 7. At the beginning of every 3 months, $450 is deposited into a savings account for 15 years. The money remained in the account for another 5 years and earned an average of 3.45% compounded monthly during the 20 year period. Calculate the accumulated amount and the interest earned in the account. 8. A grandmother opened a savings account for her granddaughter on the day she was born and deposited $1000. Each year on her birthday, she deposited $1000 for 16 years. If the account earned 4.38% compounded daily, how much was in the account on her granddaughter's 21st birthday and what is the interest earned? 9. What will be the value of a lease contract if it requires you to make lease payments of $450 at the beginning of each month for five years? Assume an interest rate of 6% compounded semi-annually. 10. What should be the amount in an RRSP that is earning 5.5% compounded quarterly if it can be converted to an RRIF that will provide $3000 at the beginning of every month for ten years? 11. Marlin secured a lease on a machine by paying $4500 as a down payment and lease payments of $750 at the beginning of every month for a period of five years. At the end of five years, he would have to pay $5000 to own the machine. If money is worth 5.55% compounded quarterly, calculate the price of the machine. 12. Nina leased a car for four years by making a down payment of $1500 and lease payments of $180 at the beginning of every month for four years. She could own the car by paying $4000 at the end of the lease. If money is worth 4% compounded quarterly, calculate the price of the car.
13. Lydia deposited $10,000 into a savings account at the beginning of every year for 10 years. The account was growing at 3% compounded monthly. After the 10 year period, she left the accumulated money in the account to grow for another year. a. What was the balance in the account at the end of 11 years? b. What was the total interest earned? 14. Chase has been contributing $1500 into a retirement fund at the beginning of each quarter for the past 15 years. He decided to stop making payments and to allow his investment to grow for another 5 years. If money can earn 8% compounded monthly, how much interest would he have earned over the 20-year period? 15. Your retirement fund currently has a balance of $40,000 and you deposit into this fund $400 at the beginning of every month starting from today for the next 15 years. Calculate the accumulated value of the fund and the amount of interest earned if the fund earns 4.25% compounded quarterly. 16. You deposit $25,000 into a savings account yielding 3.5% compounded semi-annually. In addition, you plan to deposit $1000 at the beginning of every three months for ten years into this account. Determine the total amount you will have at the end of ten years and the interest earned. 17. A contract requires lease payments of $400 at the beginning of every month for ten years. a. What is the value of the contract if the cost of borrowing is 4.5% compounded annually? b. What is the value of the contract if the cost of borrowing is 4.5% compounded daily? 18. Soong decided to retire in Canada and wants to receive a retirement income of $4000 at the beginning of every month for 15 years from her savings. If a bank in Vancouver is offering her an interest rate of 5% compounded annually, how much would she have to invest to get her planned retirement income? What is the financial benefit for her to invest in a bank in Niagara Falls if they are offering her an interest rate of 4.99% compounded quarterly?
19. How much lesser will the value of an RRSP account be at the end of ten years if you contribute $200 at the beginning of every month instead of $2400 at the beginning of every year? Assume the account earns an interest rate of 5.2 compounded semi-annually in both cases. 20. Nathan deposited $500 at the beginning of each month into a fund. His wife deposited $1500 at the beginning of every three months into a similar fund. If they continue these deposits, which fund will have a larger balance at the end of 15 years? The funds earn interest at 6.2% compounded semi-annually. 21. How much should Main Corp. pay for an annuity that would give it $20,000 at the beginning of every year for the first five years and $30,000 at the beginning of every year for the next three years, if the interest rate is 7% compounded monthly throughout the term? 22. What is the purchase price of an annuity that provides beginning-of-month payments of $400 for the first three years and $600 for the next two years? Assume that the interest rate is 4% compounded quarterly throughout the time period.
23. The monthly rent on an apartment is $1750 per month, payable at the beginning of each month. What single payment in advance on the first day of rental would be equal to three years' rent? Assume an interest rate of 4.38% compounded daily.
24. Machinery can be leased for your manufacturing plant with a payment of $4500 at the beginning of every three months. What single payment in advance on the first day of the lease would be equal to four years of lease payments? Assume an interest rate of 4.4% compounded monthly. 25. Jasmine, an office administrator, was evaluating the following quotation that she received for the purchase of a printing machine for her company:
■
Lease Option: Make payments of $750 at the beginning of every month for two years. At the end of two years pay $3000 to own the printer.■
Purchase Option: Make a payment of $19,500 immediately. The cost of borrowing is 6% compounded semi-annually. a. Which option is economically better for the company? b. In the lease option, what will be the book value of the printer at the end of one year? 26. Madison had the option of purchasing a machine for $59,000 or making lease payments of $1200 at the beginning of every month for four years and then paying $15,000 at the end of the fourth year to own it. The cost of borrowing money is 7% compounded quarterly. a. Which option is economically better for her? b. In the lease option, what will be the book value of the machine at the end of three years? 27. Ryder invested $1000 at the beginning of every three months into his RRSP account from age 35 to 45 and left the money to grow until his retirement at the age of 65. Bill invested $1000 at the beginning of every three months into a similar RRSP account from age 45 until his retirement at the age of 65. Assuming money earns 5.8% compounded annually in both RRSP accounts, calculate who had the greater accumulated value and by how much when they retired? 28. Chan made deposits of $250 into a fund at the beginning of every month from age 40 to 48. He left the money to accumulate in the fund until he reached the age of 60. David made deposits of $250 at the beginning of every month into an RRSP from age 48 until he reached the age of 60. Assuming money earns 6.25% compounded semi -annually in both cases, calculate who had the greater accumulated value and by how much when they reached 60 years of age?29. A lottery winner is offered a choice of either receiving $20,500 at the beginning of every year for ten years or receiving $18,000 at the beginning of every six months for five years. If the interest rate is 4.4% compounded quarterly, which choice is economically better (in current value) and by how much more? 30. You have a choice of either receiving $15,000 at the beginning of every year for ten years or receiving $2250 at the beginning of every month for five years. If the interest rate is 4.8% compounded semi-annually, which choice is economically better (in current value) and by how much more? 31.
32.
If the future value of an annuity, the periodic interest rate, 'i', and the number of payment periods, 'n', are known, we can use the future value formula to solve for the unknown 'PMT'. Similarly, if the present value of an annuity, the periodic interest rate, 'i', and the number of payment periods, 'n', are known, we can use the present value formula to solve for the unknown 'PMT'.Calculating the Periodic Payment of an Ordinary Simple Annuity, Given Present Value
Rodney received a $35,000 home improvement loan from his bank at an interest rate of 6% compounded monthly. (i) How much would he need to pay every month to settle the loan in ten years? (ii) What was the amount of interest charged? This is an ordinary simple annuity as:
■
Payments are made at the end of each payment period (monthly)■
Compounding period (monthly) = payment period (monthly) (i) Using Formula 10.2(b), PV = PMT -i i 1-^1+ h n ; E 35,000 = PMT . . 0 005 1-^1+0 005h-120 ; E Solving for 'PMT', we obtain, PMT = . . , 0 005 1 1 0 005 35 000 120 -^ + h -; E = . ... , 90 073453 35 000 6 @ = $388.571756... Therefore, he would need to pay $388.57 every month to settle the loan in ten years. (ii) Interest Charged over the time period = n(PMT) - PV = 120(388.57) - 35,000 = $11,628.40 Therefore, the amount of interest charged during the period of the loan would be $11,628.40.10.6
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Calculating Periodic Payment (PMT)
Example 10.6(a)
