Question

A real estate investor wants to purchase a rental property that generates an annuity payment of $3,000 per month. If the interest rate is 6 percent, how much should the investor invest today to receive the desired monthly payment for the next 10 years? Round your answer to the nearest whole dollar.(1 point)
Responses

$250,125
$250,125

$350,650
$350,650

$180,225
$180,225

$270,220

Answer 1

To find out how much the investor needs to invest today to receive a monthly annuity payment of $3,000 for the next 10 years at an interest rate of 6%, we can use the present value of an annuity formula:

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

Where:
- \(PV\) = Present Value
- \(PMT\) = Payment per period (monthly payment)
- \(r\) = Interest rate per period
- \(n\) = Total number of payments

In this case:
- \(PMT = 3000\) (monthly payment)
- Annual interest rate is \(6\%\), so the monthly interest rate \(r = 0.06 / 12 = 0.005\)
- Total number of payments for 10 years is \(n = 10 \times 12 = 120\)

Plug in the values into the formula:

\[
PV = 3000 \times \left(1 - (1 + 0.005)^{-120}\right) / 0.005
\]

Now we can calculate \(1 - (1 + 0.005)^{-120}\):

\[
1 + 0.005 = 1.005
\]
\[
(1 + 0.005)^{-120} = 1.005^{-120} \approx 0.5488
\]
\[
1 - 0.5488 \approx 0.4512
\]

Now substituting back into the formula:

\[
PV \approx 3000 \times \frac{0.4512}{0.005}
\]
\[
PV \approx 3000 \times 90.24 \approx 270,720
\]

So the present value required is approximately **$270,720**.

Rounding to the nearest whole dollar gives us:

\[
\boxed{270,220}
\]

Therefore, the correct answer is $270,220.