Question

Determine the amount needed such that when it comes time for retirement, an individual can make monthly withdraws in the amount of $2,154 for 30 years from an account paying 5.1% compounded monthly. Round your answer to the nearest cent. a. $396,721.78 b. $398,407.85 c. $775,440 d. $1,833,962.40

Answer 1

The amount is $\boxed{\ 396721.78}.$

Further explanation:

Given:

The options are as follows,

(a). $\ 396721.78$

(b). $\ 398407.85$

(c). $\ 775440$

(d). $\ 1833962.40$

Explanation:

The number of months are $360{\text{ months}}.$

The account pay interest rate is $\text{i} = 5.1\%$ compounded monthly.

The amount withdraws each month is $\ 2154.$

The value of $\text{n}$ is 360.

The value of $\text{i}$ is $5.1\%.$

The present value formula can be expressed as follows,

$\boxed{{\text{PV}} = A \times \dfrac{{\left[ {1 - {{\left( {1 + \dfrac{i}{n}} \right)}^{ - n}}} \right]}}{{\dfrac{i}{n}}}}$

The present value can be obtained as follows,

$\begin{aligned}{\text{PV}} &= {\text{A}} \times \dfrac{{\left[ {1 - {{\left( {1 + \dfrac{i}{{12}}} \right)}^{ - 360}}} \right]}}{{\dfrac{i}{{12}}}} \\&= 2154 \times \frac{{\left[ {1 - {{\left( {1 + \dfrac{{0.051}}{{12}}} \right)}^{ - 360}}} \right]}}{{\dfrac{{0.051}}{{12}}}}\\&= 2154 \times \dfrac{{\left[ {1 - 0.21724} \right]}}{{0.00425}}\\\end{aligned}$

Further solve the above equation.

$\begin{aligned}{\text{PV}} &= 2154 \times \frac{{0.78276}}{{0.00425}}\\&= 2154 \times 184.20\\&= \ 396721.78\\\end{aligned}$

The amount is $\boxed{\ 396721.78}.$

Option (a) is correct.

Option (b) is not correct.

Option (c) is not correct.

Option (d) is not correct.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Annuity

Keywords: amount, needed, retirement, individual, time of retirement, withdraws, account, paying, 5.1% compounded monthly, nearest cents, monthly, interest rate.

Answer 2

You are given a monthly withdrawal of $2,154 for 30 years from an account paying 5.1% compounded monthly. You are asked to find the initial payment. use annuity equation here.

P = A [(1+i)^n]-1/i(1+i)^n
P = $2,154 [(1+(0.051/12))^30*12] - 1 / (0.051/12)(1+(0.051/12))^30*12
P = $396,721.78

The answer is letter a.