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. a. ruthann will meet her annual goal of exactly $32,523 for retirement. b. ruthann will meet her annual goal of $32,523 for retirement with an excess of $20,791.60. c. ruthann will not make her annual goal of $32,523 and will need $1,039.85 to supplement her yearly income when she retires. d. ruthann will not make her annual goal of $32,523 and will need $10,395.80 to supplement her yearly income when she retires. please select the best answer from the choices provided

Explanation:

Step1: Calculate annual 401(k) contribution

First, find 11% of RuthAnn's annual income.
$\text{Annual Contribution} = 0.11 \times 36278.13 = 3990.5943$

Step2: Calculate years until retirement

Subtract current age from retirement age.
$\text{Years to Retirement} = 65 - 28 = 37$

Step3: Find future value of contributions

Use the future value of an ordinary annuity formula: $FV = P \times \frac{(1+r)^n - 1}{r}$, where $P=3990.5943$, $r=0.071$, $n=37$.
$$FV = 3990.5943 \times \frac{(1+0.071)^{37} - 1}{0.071}$$
First calculate $(1.071)^{37} \approx 11.3322$
Then $\frac{11.3322 - 1}{0.071} = \frac{10.3322}{0.071} \approx 145.5239$
$FV \approx 3990.5943 \times 145.5239 \approx 580734.42$

Step4: Calculate annual retirement income

Use the present value of an ordinary annuity formula to find annual withdrawal $A$: $PV = A \times \frac{1-(1+r)^{-m}}{r}$, where $PV=580734.42$, $r=0.071$, $m=30$.
Rearrange to solve for $A$: $A = PV \times \frac{r}{1-(1+r)^{-m}}$
First calculate $(1.071)^{-30} \approx 0.1237$
$\frac{0.071}{1-0.1237} = \frac{0.071}{0.8763} \approx 0.0810$
$A \approx 580734.42 \times 0.0810 \approx 31483.15$

Step5: Calculate income shortfall

Subtract actual annual income from needed income.
$\text{Shortfall} = 32523 - 31483.15 = 1039.85$

Answer:

c. RuthAnn will not make her annual goal of $32,523 and will need $1,039.85 to supplement her yearly income when she retires.