To solve the given problem of finding the future value of an annuity due, we can proceed as follows:
An annuity due requires adjusting the regular annuity formula by multiplying the entire future value by an additional factor of [tex]\( (1 + r) \)[/tex], where [tex]\( r \)[/tex] is the interest rate. This adjustment accounts for the payment being made at the beginning of each period.
Given the parameters:
- Amount of payment: [tex]$4400
- Payment frequency: Annually
- Duration: 1 year
- Annual interest rate: 5%
Step-by-step solution:
1. Identify the required components:
- Payment amount (\( PMT \)): $[/tex]4400
- Number of payments per year ([tex]\( n \)[/tex]): 1 (annually)
- Number of years ([tex]\( t \)[/tex]): 1
- Annual interest rate ([tex]\( r \)[/tex]): 5% (or 0.05 as a decimal)
2. Future Value of an Ordinary Annuity (FV) Formula:
The formula for the future value of an ordinary annuity is:
[tex]\[
FV = PMT \times \left( \frac{(1 + r)^t - 1}{r} \right)
\][/tex]
Since the number of payments is only one (annually) and the number of years is also one, for ordinary annuity:
[tex]\[
FV_{ordinary} = 4400 \times \left( \frac{(1 + 0.05)^1 - 1}{0.05} \right) = 4400 \times 1 = 4400
\][/tex]
3. Adjust for Annuity Due:
To adjust for an annuity due, you multiply the future value of the ordinary annuity by [tex]\( (1 + r) \)[/tex]:
[tex]\[
FV_{due} = FV_{ordinary} \times (1 + r)
\][/tex]
Substituting the values, we get:
[tex]\[
FV_{due} = 4400 \times (1 + 0.05) = 4400 \times 1.05 = 4620.0
\][/tex]
Hence, the future value of the annuity due is calculated as $4620.00.
