Questions: Find the present value PV of the annuity account necessary to fund the withdrawal given. (Assume end-of-period withdrawals and compounding at the nearest cent.) 2,700 per month for 10 years, if the account earns 6% per year PV=

Find the present value PV of the annuity account necessary to fund the withdrawal given. (Assume end-of-period withdrawals and compounding at the nearest cent.)
2,700 per month for 10 years, if the account earns 6% per year
PV=

Transcript text: Find the present value PV of the annuity account necessary to fund the withdrawal given. (Assume end-of-period withdrawals and compounding at the nearest cent.) $\$ 2,700$ per month for 10 years, if the account earns $6 \%$ per year $P V=\$$

Solution

Solution Steps

To find the present value (PV) of an annuity, we can use the formula for the present value of an ordinary annuity:

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

where:

$ P $ is the payment amount per period ($2,700 per month),
$ r $ is the monthly interest rate (annual rate divided by 12),
$ n $ is the total number of payments (number of years multiplied by 12).

Given:

Monthly payment $ P = 2700 $
Annual interest rate $ 6\% $ or $ 0.06 $
Number of years $ 10 $

We need to convert the annual interest rate to a monthly interest rate and calculate the total number of payments.

Step 1: Define the Variables

We are given the following values:

Monthly payment $ P = 2700 $
Annual interest rate $ r_{annual} = 0.06 $
Number of years $ t = 10 $

Step 2: Convert Annual Rate to Monthly Rate

To find the monthly interest rate, we calculate: \[ r = \frac{r_{annual}}{12} = \frac{0.06}{12} = 0.005 \]

Step 3: Calculate Total Number of Payments

The total number of payments over 10 years is: \[ n = t \times 12 = 10 \times 12 = 120 \]

Step 4: Calculate Present Value

Using the formula for the present value of an ordinary annuity: \[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] Substituting the values: \[ PV = 2700 \times \left( \frac{1 - (1 + 0.005)^{-120}}{0.005} \right) \] Calculating this gives: \[ PV \approx 243198.32 \]