In an ordinary simple annuity, payments are made at the end of each payment period and the compounding period is equal to the payment period. In this section, you will learn how to calculate the future value and present value of an ordinary simple annuity.
Future Value of an Ordinary Simple Annuity
Consider an example where Abriella decides to invest $1000 at the end of every year for five years in a savings account that earns an interest rate of 10% compounded annually. She wants to find out how much she would have at the end of the five-year period. In other words, she wants to find out the future value of her investments at the end of five years.
To calculate the future value of her investments at the end of five years, we could calculate the future value of each of her investments using the compound interest formula and then add all the future values.
From the compound interest Formula 9.1(a), we know that the future value of each payment is
FV = PV(1 + i)n
where, FV = future value, PV = present value,
i = interest rate for the compounding period (periodic interest rate), and n = total number of compounding periods for each payment.
In this example:
■
'PV' for each payment is $1000■
'i' for each payment is i =m j = . 1 0 10 = 0.10 per annum
■
'n' for each payment is not the same. 'n' for each payment starting from the 1st payment is 4, 3, 2, 1, and 0.Exhibit 10.2(a): Future Value of Ordinary Simple Annuity Payments
Sum of the future values of her investment = 1464.10 + 1331.00 + 1210.00 +1100.00 + 1000.00 = $6105.10
Therefore, if she invests $1000 every year for five years at 10% compounded annually in a savings account, she would have a total of $6105.10 at the end of five years.
10.2
|
Future Value and Present Value of an
Ordinary Simple Annuity
In compound interest,
'n' is the number of
compounding periods during the term.
The future value of an annuity is the sum of the accumulated value of each periodic payment. Individual future values calculated are in geometric series with 'n' terms, where the 1st term is PMT and the
common ratio is (1+i). By applying the formula for the sum of a geometric series you will get the simplified
In an ordinary simple annuity, payments are made at the end of each payment period and the compounding period is equal to the payment period. In this section, you will learn how to calculate the future value and present value of an ordinary simple annuity.
Future Value of an Ordinary Simple Annuity
Consider an example where Abriella decides to invest $1000 at the end of every year for five years in a savings account that earns an interest rate of 10% compounded annually. She wants to find out how much she would have at the end of the five-year period. In other words, she wants to find out the future value of her investments at the end of five years.
To calculate the future value of her investments at the end of five years, we could calculate the future value of each of her investments using the compound interest formula and then add all the future values.
From the compound interest Formula 9.1(a), we know that the future value of each payment is
FV = PV(1 + i)n
where, FV = future value, PV = present value,
i = interest rate for the compounding period (periodic interest rate), and n = total number of compounding periods for each payment.
In this example:
■
'PV' for each payment is $1000■
'i' for each payment is i =m j = . 1 0 10 = 0.10 per annum
■
'n' for each payment is not the same. 'n' for each payment starting from the 1st payment is 4, 3, 2, 1, and 0.Exhibit 10.2(a): Future Value of Ordinary Simple Annuity Payments
Sum of the future values of her investment = 1464.10 + 1331.00 + 1210.00 +1100.00 + 1000.00 = $6105.10
Therefore, if she invests $1000 every year for five years at 10% compounded annually in a savings account, she would have a total of $6105.10 at the end of five years.
10.2
|
Future Value and Present Value of an
Ordinary Simple Annuity
In compound interest,
'n' is the number of
compounding periods during the term.
The future value of an annuity is the sum of the accumulated value of each periodic payment. Individual future values calculated are in geometric series with 'n' terms, where the 1st term is PMT and the
common ratio is (1+i). By applying the formula for the sum of a geometric series you will get the simplified
'FV' Formula 10.2 (a).
Now, if there were many payments for an annuity (e.g., she invested the same amount for 15 years, compounded annually), the above method would become too time-consuming.
A simplified formula to calculate the future value of an ordinary simple annuity is given by: Future Value of an Ordinary Simple Annuity
FV = PMT
where 'n' is the number of payments during the term, 'PMT' is the amount of the periodic payment, and 'i' is the periodic interest rate.
Calculating the future value of her investment at the end of five years using this simplified formula,
n = 1 payment/year # 5 years = 5 annual payments
i = m j = . 1 0 10 = 0.10 PMT = $1000 FV = PMT i i 1+ n-1 ^ h ; E = 1000 . . 0 10 1+0 105-1 ^ h ; E = 1000 [6.1051] = $6105.10
Therefore, the future value of the investment is $6105.10.
In the calculation of the future value of annuities, the amount of interest is calculated as follows:
Amount of Interest Earned = Future Value of the Investments - Amount Invested Over the Term I = FV - n(PMT)
In this example, I = 6105.10 - 5(1000.00)
= 6105.10 - 5000.00 = $1105.10
Therefore, she would earn $1105.10 from this investment.
Calculating the Future Value, Total Investment, and Interest Earned in an Ordinary Simple Annuity
Rita invested $200 at the end of every month for 20 years into an RRSP. Assume that the interest rate was constant at 6% compounded monthly over the entire term.
(i) What was the accumulated value of the investment at the end of the term? (ii) What was the total investment over the term?
(iii) What was the amount of interest earned? This is an ordinary simple annuity as:
■
Payments are made at the end of each payment period (monthly)■
Compounding period (monthly) = payment period (monthly)Formula 10.2(a)
In an annuity, 'n' is the number of payments during the term.
Example 10.2(a)
(i) Using Formula 10.2(a),
FV = PMT 1 ii 1 n + -^ h ; E = 200 . . 0 005 1+0 005240-1 ^ h ; E = 200 [462.040895...] = $92,408.17903...
Therefore, the accumulated value of the investment at the end of the term was $92,408.18. (ii) Total Investment = $200 per month # 240 payments = $48,000.00.
Therefore, the total investment over the term was $48,000.00 (iii) Interest Earned = $92,408.18 - $48,000.00 = $44,408.18
Therefore, the amount of interest earned was $44,408.18.
Using the Financial Calculator to Solve Problems
Before you start solving problems, check and set the following in the Texas Instruments BA II Plus financial calculator:
1. Set the number of decimals to 9. (Set this only when you start using the calculator.)
Press 2ND then press FORMAT (this is the secondary
function of the decimal key).
Enter 9 then press ENTER to set the number of decimals to 9.
Press 2ND then QUIT (this is the secondary key above CPT).
1 2 3
2. Check the settings for end-of-period (ordinary annuity) or beginning-of-period (annuity due) calculations. (You need to check this before you work out problems.)
Pressing 2ND then BGN (secondary function above PMT) will display
the current setting - either END or BGN. (Select END for ordinary annuity
problems and BGN for annuity due problems.)
END would mean end-of-period setting (for an ordinary annuity). To change
this setting to beginning-of-period setting (for annuity due), press 2ND then SET (secondary function above ENTER). You can switch between END
and BGN by pressing 2ND then SET again.
If BGN setting is selected, a small BGN appears at the top-right corner of
your screen and will remain there throughout your calculations. However, if
END is selected, nothing will appear.
Once you select your setting, press 2ND then QUIT (secondary function
above CPT) and return to your calculation.
1 2 3 4 Solution continued
Solving Example 10.2(a)(i) using the Texas Instruments BA II Plus calculator, to find 'FV' Where, P/Y = 12, C/Y = 12, N = 240, I/Y = 6, PMT = $200
Calculating the Future Value, Total Investment, and Interest Earned in an Ordinary Simple Annuity Combined with a Compound Interest Period
Jack deposited $1500 into an account every three months for a period of four years. He then let the money grow for another six years without investing any more money into the account. The interest rate on the account was 6% compounded quarterly for the first four years and 9% compounded quarterly for the next six years. Calculate (i) the accumulated amount of money in the account at the end of the 10-year period and (ii) the total interest earned.
This is an ordinary simple annuity as:
■
Payments are assumed to be at the end of each payment period (quarterly)■
Compounding period (quarterly) = payment period (quarterly)Solution
Cash-Flow Sign Convention
Transaction PV FV
Investment Outflow (-) Inflow (+) Loan Inflow (+) Outflow (-)
Clear past values in the memory of the function keys.
This opens the P/Y, C/Y worksheet to set values.
Set payments per year (P/Y) equal to compounding periods per year
(C/Y) as 12. You can scroll down using the down arrow key to view C/Y,
which will be automatically set to 12.
This closes the P/Y, C/Y worksheet.
Number of payments (n). Nominal interest rate per year (j).
In annuity problems, we will usually find either FV or PV. To avoid errors
when you are finding FV set PV to ‘0’ and vice versa.
Periodic payments can be cash inflows or outflows,
so set the sign accordingly.
In this problem, it is a cash outflow (money paid out for the investment), therefore, the periodic payment is negative.
2 3 5 8 1 4 6 7 Example 10.2(b) When the payment date of the investment is not stated, it is assumed to be at the end of the payment period.
(i) Using Formula 10.2(a), FV = PMT i i 1+ n-1 ^ h ; E FV1 = 1500 1 0 0150 015.. 1 16 + -^ h = G = 1500[17.932369...] = $26,898.55477...
This 'FV1'amount now becomes the present value for the compounding period. We now need to find the future value of this amountusing the compound interest formula.
FV2 = PV2(1 + i)n
Here, 'n', the number of compounding periods = m # t = 24
FV2 = 26,898.55477...(1 + 0.0225)24 = $45,882.65566... Therefore, the accumulated amount of money in the account at the end of the 10-year period was $45,882.66.
(ii) Total Invested = n(PMT)
= 16 # 1500.00 = $24,000.00
Interest Earned = FV - n(PMT)
= 45,882.65566... - 24,000.00 = $21,882.65566...
Therefore, the total interest earned over the period was $21,882.66. Calculating the Future Value when Payment Changes
Grace saved $500 at the end of every month in an RRSP for five years and thereafter $600 at the end of every month for the next three years. If the investment was growing at 3% compounded monthly, calculate the maturity value of her RRSP at the end of eight years.
This is an ordinary simple annuity as:
■
Payments are made at the end of each payment period (monthly)■
Compounding period (monthly) = payment period (monthly)Solution
continued
Example 10.2(c)
(i) Using Formula 10.2(a), FV = PMT i i 1+ n-1 ^ h ; E FV1 = 1500 1 0 0150 015.. 1 16 + -^ h = G = 1500[17.932369...] = $26,898.55477...
This 'FV1'amount now becomes the present value for the compounding period. We now need to find the future value of this amountusing the compound interest formula.
FV2 = PV2(1 + i)n
Here, 'n', the number of compounding periods = m # t = 24
FV2 = 26,898.55477...(1 + 0.0225)24 = $45,882.65566... Therefore, the accumulated amount of money in the account at the end of the 10-year period was $45,882.66.
(ii) Total Invested = n(PMT)
= 16 # 1500.00 = $24,000.00
Interest Earned = FV - n(PMT)
= 45,882.65566... - 24,000.00 = $21,882.65566...
Therefore, the total interest earned over the period was $21,882.66. Calculating the Future Value when Payment Changes
Grace saved $500 at the end of every month in an RRSP for five years and thereafter $600 at the end of every month for the next three years. If the investment was growing at 3% compounded monthly, calculate the maturity value of her RRSP at the end of eight years.
This is an ordinary simple annuity as:
■
Payments are made at the end of each payment period (monthly)■
Compounding period (monthly) = payment period (monthly)Solution
continued
Example 10.2(c)
Solution
Calculating the future value of PMT(i) at the end of the five years
Using Formula 10.2(a),
FV = PMT 1 ii 1 n + -^ h ; E FV1 = 500 1 0 0025. -1 0.0025 + ^ h ; E 60 = 500[64.646712...] = $32,323.35631...
FV1 becomes the present value for the compounding period. We now need to find the future value of this amount using the compound interest formula.
FV2 = PV2(1 + i)n
Here 'n', the number of compounding periods = m # t = 36
FV2 = 32,323.35631...(1 + 0.0025)36 = $35,363.41325
Calculating the future value of PMT(ii) at the end of the term:
Using Formula 10.2(a),
FV = PMT 1 ii 1 n + -^ h ; E FV3 = 600 1 0 0025. -1 0.0025 + ^ h ; E 36 = 600[37.620560...] = $22,572.33619...
Maturity value of investment at the end of the term: = FV2 + FV3
= 35,363.41325 + 22,572.33619... =$57,935.74944
Therefore, the maturity value of her investment at the end of eight years is $57,935.75.
Present Value of an Ordinary Simple Annuity
Consider an example where Margaret wishes to withdraw $1000 at the end of every year for the next five years from an account that pays interest at 10% compounded annually. How much money should she deposit into this account now?
To calculate the present value of all the payments at the beginning of the five-year period, we can calculate the present value of each payment and then add all the present values using the compound interest formula as shown below:
As payments that she would be receiving in the future have to be discounted, the present value of each payment is given by the compound interest formula: PV =
i FV
1+ n
^ h = FV(1 + i)
-n
where 'n' is the number of compounding periods for each payment. 'n' for each payment starting from the 1st payment is 1,2,3,4, and 5.
Solution
Sum of Present Values of her Investment = 909.09 + 826.45 + 751.31 + 683.01 + 620.92
= $3790.79.
Now, similar to the simplified formula used in FV calculations, the simplified PV formula is given by: Present Value of an Ordinary Simple Annuity
PV = PMT
where 'n' is the number of payments during the term and 'PMT ' is the periodic payment. Calculating the present value of her payments using this simplified formula:
PV = 1000 . 0.10 1- 1+0 10-5 ^ h ; E = 1000[3.790786...] = $3790.79
Therefore, she would have to deposit $3790.79 at the beginning of the five-year period to be able to withdraw $1000 at the end of each year for five years.
In calculating the present value of an annuity, the amount of interest is calculated as follows:
Amount of Interest Earned = Amount Received over the Term - Present Value I = n(PMT) - PV
In this example, I = 5(1000.00) - $3790.79 = 5000.00 - $3790.79 = $1209.21
Therefore, she would earn interest of $1209.21.
Calculating the Present Value and Interest Earned in an Ordinary Simple Annuity
Zack purchased an annuity that provided him with payments of $1000 every month for 25 years at 5.4% compounded monthly.
(i) How much did he pay for the annuity?
(ii) What was the total amount received from the annuity and how much of this amount was the interest earned?
This is an ordinary simple annuity as:
■
Payments are assumed to be made at the end of each payment period (monthly)■
Compounding period (monthly) = payment period (monthly)The present value of an annuity is the sum of the discounted values of each periodic payment.
Exhibit 10.2(b): Present Value of Ordinary Simple Annuity Payments
Individual present values calculated are in geometric series with 'n' terms, where the 1st term is 'PMT' and the
common ratio is (1+ i ) -1. By
applying the formula for the sum of a geometric series you will get the simplified 'PV' Formula 10.2 (b).
Formula 10.2(b)
Example 10.2(d)
Sum of Present Values of her Investment = 909.09 + 826.45 + 751.31 + 683.01 + 620.92
= $3790.79.
Now, similar to the simplified formula used in FV calculations, the simplified PV formula is given by: Present Value of an Ordinary Simple Annuity
PV = PMT
where 'n' is the number of payments during the term and 'PMT ' is the periodic payment. Calculating the present value of her payments using this simplified formula:
PV = 1000 . 0.10 1- 1+0 10-5 ^ h ; E = 1000[3.790786...] = $3790.79
Therefore, she would have to deposit $3790.79 at the beginning of the five-year period to be able to withdraw $1000 at the end of each year for five years.
In calculating the present value of an annuity, the amount of interest is calculated as follows:
Amount of Interest Earned = Amount Received over the Term - Present Value I = n(PMT) - PV
In this example, I = 5(1000.00) - $3790.79 = 5000.00 - $3790.79 = $1209.21
Therefore, she would earn interest of $1209.21.
Calculating the Present Value and Interest Earned in an Ordinary Simple Annuity
Zack purchased an annuity that provided him with payments of $1000 every month for 25 years at 5.4% compounded monthly.
(i) How much did he pay for the annuity?
(ii) What was the total amount received from the annuity and how much of this amount was the interest earned?
This is an ordinary simple annuity as:
■
Payments are assumed to be made at the end of each payment period (monthly)■
Compounding period (monthly) = payment period (monthly)The present value of an annuity is the sum of the discounted values of each periodic payment.
Exhibit 10.2(b): Present Value of Ordinary Simple Annuity Payments
Individual present values calculated are in geometric series with 'n' terms, where the 1st term is 'PMT' and the
common ratio is (1+ i ) -1. By
applying the formula for the sum of a geometric series you will get the simplified 'PV' Formula 10.2 (b).
Formula 10.2(b)
Example 10.2(d)
Solution
(i) Using Formula 10.2(b),
PV = PMT 1 1i i n -^ + h -; E = 1000 . . 0 0045 1-^1+0 0045h-300 ; E = 1000 [164.438546...] = $164,438.546990...
Therefore, he paid $164,438.55 for the annuity. (ii) Amount Received = n(PMT)
= 300 # $1000.00 = $300,000.00
Interest Earned = n(PMT) - PV
= 300,000.00 - 164,438.55 = $135,561.45
Calculating the Present Value, Amount Invested, and Total Interest Charged in an Ordinary Simple Annuity
Andrew paid $20,000 as a down payment towards the purchase of a machine and received a loan for the balance amount at an interest rate of 3% compounded monthly. He settled the loan in ten years by paying $1500 at the end of every month.
(i) What was the purchase price of the machine?
(ii) What was the total amount paid to settle the loan and 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)Solution
continued
Example 10.2(e)
(i) Using Formula 10.2(b), PV = PMT -1 1 i i n -^ + h ; E = 1500 . . 0 0025 1-^1+0 0025h-120 ; E = 1500[103.561753...] = 155,342.6296... = $155,342.63 As he paid $20,000.00 as down payment for the machine:
Purchase Price = Down Payment + PV of All Payments
= 20,000.00 + 155,342.63 = $175,342.63 Therefore, the purchase price of the machine was $175,342.63. (ii) Calculating the total amount paid and the interest amount Amount Paid = n(PMT) = 120 # 1500.00 = $180,000.00 Interest Charged = n(PMT) - PV = 180,000.00 - 155,342.63 = $24,657.37
Therefore, the total amount paid to settle the loan was $180,000.00 and the amount of interest charged was $24,657.37.
Calculating the Present Value when the Interest Rate Changes
How much should Halifax Steel Inc. invest today in a fund to be able to withdraw $15,000 at the end of every three months for a period of six years? The money in the fund is expected to grow at 4.8% compounded quarterly for the first two years and 5.6% compounded quarterly for the next four years.
This is an ordinary simple annuity as:
■
Withdrawals are made at the end of each payment period (quarterly)■
Compounding period (quarterly) = payment period (quarterly)Solution
continued
Example 10.2(f)
(i) Using Formula 10.2(b), PV = PMT -1 1 i i n -^ + h ; E = 1500 . . 0 0025 1-^1+0 0025h-120 ; E = 1500[103.561753...] = 155,342.6296... = $155,342.63 As he paid $20,000.00 as down payment for the machine:
Purchase Price = Down Payment + PV of All Payments
= 20,000.00 + 155,342.63 = $175,342.63 Therefore, the purchase price of the machine was $175,342.63. (ii) Calculating the total amount paid and the interest amount Amount Paid = n(PMT) = 120 # 1500.00 = $180,000.00 Interest Charged = n(PMT) - PV = 180,000.00 - 155,342.63 = $24,657.37
Therefore, the total amount paid to settle the loan was $180,000.00 and the amount of interest charged was $24,657.37.
Calculating the Present Value when the Interest Rate Changes
How much should Halifax Steel Inc. invest today in a fund to be able to withdraw $15,000 at the end of every three months for a period of six years? The money in the fund is expected to grow at 4.8% compounded quarterly for the first two years and 5.6% compounded quarterly for the next four years.
This is an ordinary simple annuity as:
■
Withdrawals are made at the end of each payment period (quarterly)■
Compounding period (quarterly) = payment period (quarterly)Solution continued Example 10.2(f) Solution Calculating PV when interest rate is 4.8% compounded quarterly Using Formula 10.2(b), PV = PMT -1 1 i i n -^ + h ; E PV1 = 15,000 0.012 1 - (1 + 0.012) ; -8E = 15,000[7.584725...] = $113,770.8865... Calculating PV when interest rate is 5.6% compounded quarterly Using Formula 10.2(b), PV = PMT -1 1 i i n -^ + h ; E PV2 = 15,000 0.014 1 - (1 + 0.014) ; -16E = 15,000[14.245867...] = $213,668.0186...
PV2 becomes the future value for the compounded period. We now need to find the present value of that amount using the compound interest formula.
PV3 = FV3(1 + i)-n
= 213,688.0186...(1 + 0.012)-8 = $194,238.8384...
Initial value of the investment
= PV1 + PV3
= 113,770.8865... + 194,238.8384... = 308,009.7249
Therefore, in order to be able to withdraw $15,000 at the end of every three months for six years, Halifax Steel Inc. should invest $308,009.72 today.
Solution
continued
1. Calculate the future value of each of the following ordinary simple annuities:
Periodic Payment Payment Period Term of Annuity Interest Rate Compounding Frequency
a. $2500 Every year 10 years 4.50% Annually
b. $1750 Every 6 months 7.5 years 5.10% Semi-annually
c. $900 Every 3 months 5 years 4.60% Quarterly
d. $475 Every month 4.5 years 6.00% Monthly
2. Calculate the future value of each of the following ordinary simple annuities:
Periodic Payment Payment Period Term of Annuity Interest Rate Compounding Frequency
a. $4500 Every year 12 years 4.75% Annually
b. $2250 Every 6 months 6 years 5.00% Semi-annually
c. $800 Every 3 months 8.5 years 4.80% Quarterly
d. $350 Every month 10 years 3 months 5.76% Monthly
3. Calculate the present value of each of the ordinary simple annuities in Problem 1. 4. Calculate the present value of each of the ordinary simple annuities in Problem 2.
5. Aliana saved $50 at the end of every month in her savings account at 6% compounded monthly for five years. a. What is the accumulated value of the money at the end of five years?
b. What is the interest earned?
6. Sharleen contributed $400 towards an RRSP at the end of every month for four years at 2.5% compounded monthly. a. What is the accumulated value of the money at the end of four years?
b. What is the interest amount earned?
7. Shanelle saves $600 at the end of every month in an RESP at 4.5% compounded monthly for 15 years for her child's education.
a. How much will she have at the end of 15 years?
b. If she leaves the accumulated money in the savings account for another two years, earning the same interest rate, how much will she have at the end of the period?
8. Lue makes deposits of $250 at the end of every month for ten years in a savings account at 3.5% compounded monthly. a. How much will he have at the end of ten years?
b. If he plans to leave the accumulated amount in the account for another five years at the same interest rate, how much will he have at the end of the period?
9. Carrie saved $750 of her salary at the end of every month in an RRSP earning 4% compounded monthly for 20 years. How much more would she have earned if she had saved this amount in an RRSP that was earning an interest rate of 4.25% compounded monthly?
10. Adrian invests $500 at the end of every three months in a savings account at 6% compounded quarterly for 7 years and 9 months. How much more would he have earned if he had saved it in a fund that was providing an interest rate of 6.5% compounded quarterly?
11. What is the discounted value of the following stream of payments: $3000 received at the end of every 3 months for 10 years and 6 months at 3% compounded quarterly?
12. What is the discounted value of the following stream of payments: $1250 received at the end of every month for 3 years and 2 months? Assume that money is worth 2.75% compounded monthly.
13. How much should Cortland have in a savings account that is earning 4% compounded monthly if he plans to withdraw $1500 from the account at the end of every month for ten years?
14. Calculate the amount of money that Chin Ho should deposit in an investment account that is growing at 6% compounded monthly to be able to withdraw $700 at the end of every month for four years.
15. Adler took a loan from a bank at 7% compounded monthly to purchase a car. He was required to pay the bank $300 at the end of every month for the next three years. What was the cash price of the car?
16. What would be the purchase price of an annuity that provides $500 at the end of every month for five years and earns an interest rate of 4% compounded monthly?
17. Calculate the accumulated value of annuity contributions of $500 at the end of every month for five years followed by contributions of $750 at the end of every month for the next four years if money is worth 4.2% compounded monthly. 18. Kumar invested $1000 at the end of every six months for six years followed by $1250 at the end of every six
months for the next three years into a fund that earns 3.25% compounded semi-annually. Calculate the accumulated amount at the end of nine years.
19. Donovan contributed $900 at the end of every three months for seven years into an RRSP fund that earned interest at 3.9% compounded quarterly for the first four years and 3.8% compounded quarterly for the next three years. Calculate the accumulated value of the contribution at the end of seven years and the amount of interest earned.
20. Calculate the accumulated value and the amount of interest earned on deposits of $125 made at the end of every month into an RESP fund for 12 years if the fund earned interest at 3.75% compounded monthly for 7 years and 4.35% compounded monthly for the next 5 years.
21. Jordan invested $1500 at the end of every six months for four years and then $750 at the end of every three months for the next two years. The investment earned interest at 5% compounded semi-annually the first four years and 4.8% compounded quarterly for the next two years. Calculate the accumulated value at the end of six years and the amount of interest earned.
22. Calculate the accumulated value of an annuity with payments of $1500 at the end of every three months for three years at 4.1% compounded quarterly and $1750 at the end of every three months for the next two years at 4.25% compounded quarterly.
23. Falco Inc. paid $25,000 as a down payment for a machine. The balance amount was financed with a loan at 3.25% compounded semi-annually, which required a payment of $9000 at the end of every six months for five years to settle the loan.
a. What was the purchase price of the machine? b. What was the total amount of interest charged? 24. Brandon purchased a computer-controlled machine for his machine shop by paying a down payment of $17,500.
He financed the balance amount with a loan at 4.75% compounded semi-annually, which required a payment of $10,000 at the end of every six months for three years.
a. What was the purchase price of the machine? b. What was the total amount of interest charged? 25. How much would you have to pay now for a retirement annuity that would provide $3000 at the end of every three
months at 5% compounded quarterly for the first ten years and $2500 at the end of each month at 6% compounded monthly for the following five years?
26. How much should Drake pay today for a retirement annuity that would provide him with $4000 at the end of every month for five years at 3.5% compounded monthly and $20,000 every six months for the next ten years at 4% compounded semi-annually?
27. Kayla wanted to purchase a storage locker for $5000 at her apartment building. She could either pay the entire amount or take a loan from the bank for this amount at 6.5% compounded monthly. She would have to pay $150 every month to settle the loan in four years. Which option should she choose and why?
28. Ronald and Jill were wondering if they should pay $30,000 for a parking space that was for sale in their condominium building or take a loan from the bank at 4% compounded monthly for five years, paying monthly repayment amounts of $460. Which option should they choose and why?
