a.
Use the appropriate formula to determine the periodic deposit.
b.
How much of the financial goal comes from deposits and how much comes from interest?
Periodic Deposit
Rate
Time
Financial Goal
$? at the end of every three months
3.5
%
compounded quarterly
6
years
$20 comma 000
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a. To determine the periodic deposit, you need to use the future value of an ordinary annuity formula, which is:
PMT = FV / [(1 + r/n)^(nt) - 1] * (n/r)
where:
PMT is the periodic deposit,
FV is the financial goal, which is $20,000,
r is the annual interest rate, which is 3.5% or 0.035,
n is the number of times interest applied per time period, which is 4 since it is compounded quarterly,
t is the time the money is invested for, which is 6 years.
Therefore the periodic deposit is:
PMT = 20000 / [(1 + 0.035/4)^(4*6) - 1] * (4/0.035)
PMT = 20000 / [1.92132 - 1] * (4/0.035)
PMT = 20000 / 0.92132 * 114.28571
PMT = $223.32 per quarter
b. To find out how much of the financial goal comes from deposits and how much comes from interest, you need to multiply the periodic deposit by the total number of periods, and then subtract this from the total financial goal.
Total amount from deposits = PMT * n * t
Total amount from deposits = $223.32 * 4 * 6 = $5363.97
Therefore, the amount that comes from interest is the financial goal minus the total deposits:
Total amount from interest = Financial goal - Total deposits
Total amount from interest = $20000 - $5363.97 = $14636.03
So $5363.97 comes from deposits and $14636.03 comes from interest.
Suppose that you earned a bachelor's degree and now you're teaching high school. The school district offers teachers the opportunity to take a year off to earn a master's degree. To achieve this goal, you deposit $ 4000
at the end of each year in an annuity that pays 4.5 %
compounded annually.
a.
How much will you have saved at the end of five years?
b.
Find the interest.
a. To determine how much you will have saved at the end of five years, you can use the future value of an ordinary annuity formula. The formula is:
FV = P * [(1 + r)^t - 1] / r
where:
FV is the future value,
P is the periodic deposit, which is $4000,
r is the annual interest rate, which is 4.5% or 0.045,
t is the time period, which is 5 years.
Therefore, the amount you will have saved at the end of five years is:
FV = 4000 * [(1 + 0.045)^5 - 1] / 0.045
FV = 4000 * [1.2346 - 1] / 0.045
FV = 4000 * 0.2346 / 0.045
FV = $ 20,493.33
b. To find the interest earned, you need to subtract the total deposits from the future value. So:
Interest = FV - Total deposits
Interest = $ 20,493.33 - (4000 * 5)
Interest = $ 20,493.33 - $ 20,000
Interest = $ 493.33
Therefore, the interest earned over five years is $493.33.
To determine the periodic deposit, you can use the formula:
P = (FV * (r / n)) / ((1 + (r / n)) ^ (n * t) - 1)
where:
P = periodic deposit
FV = financial goal
r = interest rate (in decimal form)
n = number of compounding periods per year
t = time (in years)
a. Using the given information, let's calculate the periodic deposit:
P = ($20,000 * (0.035 / 4)) / ((1 + (0.035 / 4)) ^ (4 * 6) - 1)
Simplifying the equation:
P = ($20,000 * 0.00875) / ((1 + 0.00875) ^ 24 - 1)
P = $175 / (1.00875 ^ 24 - 1)
Using a calculator to evaluate ((1.00875 ^ 24 - 1) = 0.225725
P = $175 / 0.225725
P ≈ $775.17
Therefore, the periodic deposit required is approximately $775.17.
b. To determine how much of the financial goal comes from deposits and how much comes from interest, we need to calculate the total amount contributed through deposits and interest separately.
Total Deposit Amount = P * (n * t)
Total Deposit Amount = $775.17 * (4 * 6)
Total Deposit Amount = $775.17 * 24
Total Deposit Amount ≈ $18,604.08
Total Interest Earned = FV - Total Deposit Amount
Total Interest Earned = $20,000 - $18,604.08
Total Interest Earned ≈ $1,395.92
Therefore, approximately $18,604.08 comes from deposits and approximately $1,395.92 comes from interest.