Janet Woo decided to retire to Florida in 7 years. What amount should Janet invest today so she can withdraw $53,500 at the end of each year for 20 years after she retires? Assume Janet can invest money at 7% compounded annually. (Do not round intermediate calculations. Round your answer to the nearest cent.)

Explanation

PV annity table:

20 years × 1 = 20 periods



7% = 7% (Table 13.2)
1

$53,500 × 10.5940 = $566,779.00

PV table:

7 years × 1 = 7 periods



7% = 7% (Table 12.3)
1

$566,779.00 × 0.6227 = $352,933.28

Well, Janet is certainly planning ahead for her retirement in sunny Florida! Let's calculate how much she needs to invest today to enjoy those $53,500 withdrawals for 20 years.

To do that, we'll use the formula for the future value of an annuity:

FV = P * [(1 + r)^n - 1] / r

Where:
FV = Future value (total amount withdrawn over 20 years)
P = Amount to be invested today
r = Interest rate (7% = 0.07)
n = Number of periods (in this case, 20 years)

Plugging in the values:

$53,500 = P * [(1 + 0.07)^20 - 1] / 0.07

Now, let's solve for P, which represents the amount Janet needs to invest today.

P = $53,500 * 0.07 / [(1 + 0.07)^20 - 1]

Calculating that out, we find that Janet needs to invest approximately $473,414.54 today to fund her retirement in sunny Florida. Now that's a sunny investment!

To find out how much Janet should invest today, we can use the formula for the present value of an annuity:

\[PV = PMT \times \frac{1 - (1+r)^{-n}}{r}\]

Where:
PV = Present Value
PMT = Payment per period
r = interest rate per period
n = number of periods

Given:
PMT = $53,500 (payment at the end of each year)
r = 7% compounded annually (0.07)
n = 20 years

Plugging in the values:

\[PV = $53,500 \times \frac{1 - (1+0.07)^{-20}}{0.07}\]

Calculating the expression inside the brackets first:

\[PV = $53,500 \times \frac{1 - (1.07)^{-20}}{0.07}\]

Calculating (1.07)^-20:

\[PV = $53,500 \times \frac{1 - 0.161}{0.07}\]

Performing the subtraction inside the bracket:

\[PV = $53,500 \times \frac{0.839}{0.07}\]

Calculating the expression:

\[PV = $53,500 \times 11.9857\]

\[PV \approx $642,265.00\]

Therefore, Janet should invest approximately $642,265.00 today in order to withdraw $53,500 at the end of each year for 20 years after she retires.

In order to determine the amount that Janet should invest today, we need to calculate the present value of an annuity.

An annuity is a series of equal payments received or made at equal intervals over a specified period of time. In this case, Janet wants to withdraw $53,500 at the end of each year for 20 years after she retires.

To calculate the present value of this annuity, we will use the formula:

PV = PMT * [(1 - (1 + r)^(-n))/r]

Where:
PV = Present Value (the amount Janet needs to invest today)
PMT = Payment per period ($53,500)
r = Interest rate per period (7% or 0.07)
n = Number of periods (20 years)

Using this formula, we can plug in the values:

PV = $53,500 * [(1 - (1 + 0.07)^(-20))/0.07]

Now let's calculate this:

PV = $53,500 * [(1 - (1.07)^(-20))/0.07]
= $53,500 * [1 - 0.2999]/0.07
= $53,500 * 14.2867
≈ $764,765.45

Therefore, Janet should invest approximately $764,765.45 today so she can withdraw $53,500 at the end of each year for 20 years after she retires.