Question

RuthAnn is 28 years old and is retiring at the age of 65. When she retires, she estimates that she will need an annual income of $32,523 for 30 years. If RuthAnn contributes 11% of her annual income to a 401(k) paying 7.1% compounded annually, will she reach her goal for retirement given that her annual income is $36,278.13? If she does not make her goal then state by what amount she will need to supplement her income. Round all answers to the nearest cent.

Answer 1

Answer:

Yes

Explanation:

From her current age of 28 to her retirement age of 65, RuthAnn has (65 - 28 =) 37 more years to work.

If she saves 11% of her annual income of $36,278.13 into a 401(k), she will be setting aside (11% * 36,278.13 =) $3,990.59 into the 401(k) account annually.

At 7.1% compounding rate, in 37 years, RuthAnn would have set aside an amount estimated by the future value of an annuity formula.

$FV = \frac{A(1+r)^{n} - 1}{r}$

where FV is the future value, the amount that would have been set aside,

A = is the annual savings,

r = is the compounding rate, and

n = is the number of years.

Therefore, the total amount that would be saved up after 37 years =

$FV = \frac{3,990.59(1+0.071)^{37} - 1}{0.071}$

= (3,990.59 * 11.6535)/0.071

= $654,990.31.

By spending $32,523 annually from an account earning 7.1% compound interest rate for 30 years, the present value of the total amount needed by RuthAnn today that will be sufficient for her retirement spending can be estimated using the present value of an annuity formula.

$PV = \frac{A(1 - (1+r)^{-n}}{r}$

= $PV = \frac{32,523(1 - (1.071)^{-30}}{0.071}$

= (32523 * 0.8723)/0.071

= $399,574.83.

Since the amount saved up ($654,990.31) is more than the total amount required for RuthAnn's retirement ($399,574.83), RuthAnn has more than sufficient to meet her Retirement goal.

Specifically, the amount she has saved up can support a maximum annual spending which can be estimated from the present value of an annuity formula.

$PV = \frac{A(1 - (1+r)^{-n}}{r}$

where PV = the amount saved up, $654,990.31,

A = the annual spending which we are estimating,

r = the 7.1% compound interest rate,

n = the number of years to retirement.

$654,990.31 = \frac{A(1 - (1.071)^{-30}}{0.071}$

= 654,990.31 = (A * 0.8723)/0.071

= A = 654,990.31/0.8723 * 0.071

= A = 53,312.29

Thus, the amount saved up can support a maximum retirement spending of $53,312.29, which is higher than the $32,523 annual income needed by RuthAnn for her retirement.