Question

Jason wants to have $30,000 accumulated at the end of 3 years and uses two funds to accumulate the $30,000. One fund provides an annual rate of return of i=.09 but has a limit of $5,000 that can be deposited each year. Another fund provides a rate of return of 4.50% per year convertible semiannually ( 2 times per year) and has no limit on the amount that can be deposited each year. On the first day of each of the next 3 years, Jason deposits $5,000 in the fund that earns interest at i= .09 , and deposits $X in the fund that earns interest at 4.50% per year convertible semiannually. Find X such that the sum of the accumulated values in the two funds at the end of 3 years is $30,000. 5) If i=.065, find the present value on 1/1/2023 of a perpetuity of $50,000 payable on 1/1 of every year continuing forever if: a) The first payment is on 1/1/2024 b) The first payment is on 1/1/2029

Answer 1

To find the value of X that will result in a sum of $30,000 at the end of 3 years, we can use the formula for compound interest. By calculating the accumulated value in each fund and setting their sum equal to $30,000, we can solve for X. The value of X is approximately $9,340.20.

Step-by-step explanation:

To determine the value of X that will result in a sum of $30,000 at the end of 3 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Let's first calculate the accumulated value in the fund that earns interest at 4.50% per year convertible semiannually:

A = P(1 + r/n)^(nt)

A = X(1 + 0.045/2)^(2*3)

A = X(1.0225)^(6)

Next, let's calculate the accumulated value in the fund that earns interest at 0.09:

A = P(1 + r/n)^(nt)

A = $5,000(1 + 0.09/1)^(1*3)

A = $5,000(1.09)^3

The sum of the accumulated values in the two funds will be $30,000:

X(1.0225)^6 + $5,000(1.09)^3 = $30,000

We can solve this equation to find X:

X(1.0225)^6 = $30,000 - $5,000(1.09)^3

X = ($30,000 - $5,000(1.09)^3) / (1.0225^6)

Calculating this using a calculator, we find that X is approximately $9,340.20.