Question

You are planning to invest in a Roth IRA using a very secure mutual fund that has paid 7% for the past 20 years and is projected to pay at least 7% for the foreseeable future. It is your plan to invest $3000 per year for the next 20 years. At the end of the first 20 years, you will let the Roth IRA fund grow without any more contributions for an additional 20 years. It is your plan to draw an annuity for an additional 25 years.

1. Solve for the cash value of the Roth IRA at the end of the first 20 years.

2. Solve for the cash value of the Roth IRA at the end of second 20 years.

3. Solve for the cash value of the annual annuity for the remaining 25 years?

Answer 1

1]

Future value of annuity = P * [(1 + r)n - 1] / r,

where P = periodic payment. This is $3000

r = periodic rate of interest. This is 7%

n = number of periods. This is 20

Future value of annuity = $3000 * [(1 + 7%)20 - 1] / 7%

Future value of annuity = $122,986.48

Value of the Roth IRA at the end of the first 20 years = $122,986.48

2]

Future value = present value * (1 + interest rate)number of years

Future value = $122,986.48 * (1 + 7%)20

Future value = $475,918.86

Value of the Roth IRA at the end of the second 20 years = $475,918.86

3]

PV of annuity = P * [1 - (1 + r)-n] / r,

where P = periodic payment. We need to calculate this.

r = interest rate per period. This is 7%.

n = number of periods. This is 25.

$475,918.86 = P * [1 - (1 + 7%)-25] / 7%

P = ($475,918.86 * 7%) / [1 - (1 + 7%)-25]

P = $40,838.84

Value of the annual annuity for the remaining 25 years is $40,838.84