Questions: Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period. 14,000; quarterly payments for 11 years; interest rate 5.9% The payment should be

Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period. 14,000; quarterly payments for 11 years; interest rate 5.9%

The payment should be

Transcript text: Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period. $\$ 14,000$; quarterly payments for 11 years; interest rate $5.9 \%$ The payment should be $\$$ $\square$ (Do not round until the final answer. Then round to the nearest cent as needed.)

Solution

Solution Steps

To find the payment for an annuity due given its future value, we can use the formula for the future value of an annuity due. The formula is:

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

Where:

$ FV $ is the future value of the annuity.
$ P $ is the payment per period.
$ r $ is the interest rate per period.
$ n $ is the total number of periods.

We need to solve for $ P $, the payment per period. Rearrange the formula to solve for $ P $:

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

Given:

$ FV = 14000 $
Quarterly payments for 11 years, so $ n = 11 \times 4 = 44 $
Annual interest rate is $ 5.9\% $, so the quarterly interest rate $ r = \frac{5.9}{4 \times 100} $

Step 1: Given Values

We are given the future value of the annuity due as $ FV = 14000 $, the annual interest rate as $ 5.9\% $, and the duration of the annuity as $ 11 $ years with quarterly payments.

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

The total number of compounding periods is calculated as: \[ n = 11 \times 4 = 44 \] The quarterly interest rate is: \[ r = \frac{5.9}{4 \times 100} = 0.01475 \]

Step 3: Calculate the Payment per Period

Using the formula for the payment $ P $ of an annuity due: \[ P = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r)} \] Substituting the known values: \[ P = \frac{14000}{\left( \frac{(1 + 0.01475)^{44} - 1}{0.01475} \right) \times (1 + 0.01475)} \] After performing the calculations, we find: \[ P \approx 224.97 \]