Adam deposited $10,000 in a 5-year CD which pays 4.2% compounded monthly

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Mathematics

Description

For each problem, show the formula used and all calculations used.

1. Adam deposited $10,000 in a 5-year CD which pays 4.2% compounded monthly. Find the account value at the end of the 5th year and the amount of interest earned during this 5 year period.

2. Bob wants to have $20,000 available in 3 years. How much should he deposit now if he can get an interest rate of 3.2% compounded quarterly and how much interest will he get?

3 .David deposits $500 at the end of each month for 7 years in an account paying 3.6% compounded monthly. He then puts the total amount on deposit in another account paying 4%interest compounded semiannually for another 8 years. Find the final amount on the deposit after the entire 15 year period.

4. Charles needs $15000 in 5 years. How much must he deposit at the end of each month for 5 years in an account paying 3.9% compounded monthly so that he will have $15000 in 5 years.

5. A $100,000 loan is taken out at 8.22% compounded monthly for 30 years for the purchase of a house. The loan requires monthly mortgage payments.

a) Find the amount of each payment

b)Find the total interest paid over the life of the loan


c)Prepare an amortization schedule for the first 3 months of this mortgage.

payment # / monthly payment / interest period / portion to principal / loan balance

0

1

2

3

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Explanation & Answer

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Question 1
Adam deposited $10,000 in a 5-year CD which pays 4.2% compounded monthly. Find the
account value at the end of the 5th year and the amount of interest earned during this 5 year period.
Solution
The future value at the end of a period of a fixed sum that is compounded periodically is:
𝑟 𝑛𝑘
𝐹𝑉 = 𝑃𝑉 (1 + )
𝑘
Where: PV = Present Value of fixed sum or deposit
r = annual percentage rate
k = number of compounding periods in one year
n = number of years
For the given question,
PV = $10,000

4.2

r = 4.2% = 100 = 0.042

k = 12 (i. e monthly compounding)

n = 5 years.
0.042 5×12
∴ FV = 10,000 × (1 +
)
12
FV = 10000(1 + 0.0035)60
FV = 10000(1.0035)60 = 10000 × 1.23322582
FV = $12,332.26
The account value at the end of the 5th year is $𝟏𝟐, 𝟐𝟑𝟑. 𝟐𝟔

Amount of Interest earned = FV − PV
= 12,332.26 − 10,000
= $2332.26
The amount of interest earned during the 5 year period is $𝟐𝟑𝟑𝟐. 𝟐𝟔

Question 2
Bob wants to have $20,000 available in 3 years. How much should he deposit now if he can get
an interest rate of 3.2% compounded quarterly and how much interest will he get?
Solution
The present value at the beginning of a period of a fixed sum a period that is compounded
periodically is:
𝑃𝑉 =

𝐹𝑉
𝑟 𝑛𝑘
(1 + )
𝑘

Where: FV = Future Value of fixed sum or deposit
r = annual percentage rate
k = number of compounding periods in one year
n = number of years
For the question,
FV = $20,000

3.2

r = 3.2% = 100 = 0.032

k = 4 (i. e quarterly compounding)

n = 3 years.
∴ PV =

20000
(1 +

𝑃𝑉 =

0.032 3×4
4 )

20000
20000
20000
=
=
12
12
(1 + 0.008)
1.008
1.10033869

𝑃𝑉 = $18,176.22
The amount Bob should deposit now is $𝟏𝟖, 𝟏𝟕𝟔. 𝟐𝟐

Amount of Interest earned = FV − PV
= 20,000 − 18,176.22
= $1,823.78
The amount of interest Bob would get in the 3 year period is $𝟏, 𝟖𝟐𝟑. 𝟕𝟖.

Question 3
David deposits $500 at the end of each month for 7 years in an account paying 3.6% compounded
monthly. He then puts the total amount on deposit in another account paying 4% interest
compounded semiannually for another 8 years. Find the final amount on the deposit after the entire
15 year period.
Solution
This problem has two parts:
Part 1: Periodic payments
The Future Value of periodic payments at the end of each period is given by:
𝑟 𝑛𝑘
(1 + ) − 1
𝑟 𝑛𝑘
𝑘
𝐹𝑉 = 𝑃𝑉 (1 + ) + 𝑃𝑀𝑇 ∙
𝑘
𝑟/𝑘
Where: PV = Present Value of or initial deposit
r = annual percentage rate
k = number of compounding periods in one year
n = number of years
𝑃𝑀𝑇 = 𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑎𝑚𝑜𝑢𝑛𝑡
PV = $0

PMT = $500

k = 12 (i. e monthly compounding)

∴ FV1 = 0 (1 +

0.036
)
12

FV1 = 0 + 500 ×

7×12

3.6

r = 3.6% = 100 = 0.036
n = 7 years.

0.036 7×12
(1 + 12 )
−1
+ 500 ∙
0.036/12

(1.003)84 − 1
1.2861108 − 1
0.2861108
= 500 ×
= 500 ×
0.003
0.003
0.003

FV1 = 500 × 95.370255
FV1 = $47,685.13

Part2: Fixed sum
The future value at the end of a period of a fixed sum that is compounded periodically is:
𝑟 𝑛𝑘
𝐹𝑉 = 𝑃𝑉 (1 + )
𝑘

The Future Value, 𝐹𝑉1 realized in part 1, ...


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