Mary needs $9000 in 7 years. What amount can she deposit at the end of each quarter at 6% interest compounded quarterly so she will have her $9000?
what is the future value of $500 in a bank account for 10 years at 10% percent compounded bimonthly
To determine the amount Mary needs to deposit at the end of each quarter, we can use the formula for future value of a series of deposits:
Future Value = Payment * [(1 + r)^n - 1] / r
Where:
Payment = amount to be deposited at the end of each quarter
r = interest rate per period
n = number of periods
In this case, Mary needs to deposit at the end of each quarter for a total of 7 years, which is equivalent to 28 quarters (since there are 4 quarters in a year). The interest rate per period is 6% divided by 4 (as it is compounded quarterly).
Let's plug in the values and solve for the Payment:
$9000 = Payment * [(1 + (0.06 / 4))^28 - 1] / (0.06 / 4)
Simplifying the equation:
$9000 = Payment * (1.015)^28 - 1 / 0.015
To solve for Payment, we rearrange the equation:
Payment = $9000 * (0.015) / [(1.015)^28 - 1]
Using a calculator, we can evaluate the right-hand side of the equation:
Payment ≈ $9000 * 0.015 / (1.448105 - 1)
Payment ≈ $9000 * 0.015 / 0.448105
Payment ≈ $300
Therefore, Mary needs to deposit approximately $300 at the end of each quarter to have $9000 after 7 years.
To calculate the amount that Mary needs to deposit at the end of each quarter, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
- A is the final amount (in this case, $9000)
- P is the principal (the amount Mary needs to deposit at the end of each quarter)
- r is the annual interest rate (6% or 0.06 as a decimal)
- n is the number of times interest is compounded per year (quarterly compounding, so n = 4)
- t is the number of years (7 years)
We need to solve for P in this formula. Rearranging the formula, we have:
P = A / (1 + r/n)^(nt)
Substituting the given values into the equation, we can solve for P:
P = 9000 / (1 + 0.06/4)^(4*7)
Using a calculator, we can simplify and evaluate the expression inside the parentheses first:
P = 9000 / (1 + 0.015)^(28)
P = 9000 / (1.015)^(28)
Now, we can calculate the final amount:
P ≈ 9000 / 1.468424
P ≈ 6132.87
Therefore, Mary would need to deposit approximately $6132.87 at the end of each quarter to have $9000 in 7 years.
Part I:
As a financial planner a client comes to you for investment advice. After meeting with him and understanding his needs, you offer him the following two investment options:
Option 1 (refer to section on Mathematics of Finance in your text.): Invest $23,000 in a savings account at 4.25% interest compounded quarterly.
Option 2 (refer to section on Mathematics of Finance in your text): Invest into an ordinary annuity where $5,000 is deposited each year into an account that earns 6.6% interest compounded annually.
SPREADSHEET:
Set up the formula for compound interest for Option 1 and the formula for Future Value of an Annuity for Option 2 in an Excel spreadsheet to calculate the amount earned at the end of 5 years.