Questions: Find the periodic withdrawals PMT for the given annuity account. (Assume end-of-period withdrawals and compounding at the same intervals as withdrawals. Round your answer to the nearest cent.) 200,000 at 2%, paid out quarterly for 19 years PMT =

Find the periodic withdrawals PMT for the given annuity account. (Assume end-of-period withdrawals and compounding at the same intervals as withdrawals. Round your answer to the nearest cent.)
200,000 at 2%, paid out quarterly for 19 years
PMT =

Transcript text: Find the periodic withdrawals PMT for the given annuity account. (Assume end-of-period withdrawals and compounding at the same intervals as withdrawals. Round your answer to the nearest cent.) $200,000 at 2%, paid out quarterly for 19 years PMT $=\$$

Solution

Solution Steps

To find the periodic withdrawals (PMT) for an annuity, we can use the formula for the present value of an annuity. The formula is:

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

Where:

$ PV $ is the present value of the annuity ($200,000 in this case).
$ r $ is the interest rate per period (annual rate divided by the number of periods per year).
$ n $ is the total number of periods (years multiplied by the number of periods per year).
$ PMT $ is the periodic payment we need to find.

Rearrange the formula to solve for PMT:

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

Step 1: Given Values

We are given the following values for the annuity:

Present Value ($ PV $) = \$200,000
Annual Interest Rate = $ 0.02 $
Compounding Periods per Year = $ 4 $ (quarterly)
Duration in Years = $ 19 $

Step 2: Calculate Interest Rate per Period and Total Number of Periods

The interest rate per period ($ r $) is calculated as: \[ r = \frac{0.02}{4} = 0.005 \] The total number of periods ($ n $) is: \[ n = 19 \times 4 = 76 \]

Step 3: Calculate Periodic Payment (PMT)

Using the formula for the periodic payment: \[ PMT = PV \times \frac{r}{1 - (1 + r)^{-n}} \] Substituting the known values: \[ PMT = 200000 \times \frac{0.005}{1 - (1 + 0.005)^{-76}} \] Calculating this gives: \[ PMT \approx 3169.66 \]