Answer:
$15,741.05
Step-by-step explanation:
To find how much should be deposited now to yield a series of future annuity payments, we can use the Present Value of an Annuity Due formula:
[tex]PV=\dfrac{PMT\left(1-\dfrac{1}{\left(1+\dfrac{r}{n}\right)^{nt}}\right)\left(1+\dfrac{r}{n}\right)}{\dfrac{r}{n}}[/tex]
where:
- PV = Future Value
- PMT = Payment Amount
- r = interest rate per year (decimal form)
- n = number of times interest is applied per year
- t = time in years
In this case:
- PMT = $700
- r = 6% = 0.06
- n = 2 (semi-annually)
- t = 18 years
Substitute the given values into the formula and solve for PV:
[tex]PV=\dfrac{700\left(1-\dfrac{1}{\left(1+\dfrac{0.06}{2}\right)^{2\cdot 18}}\right)\left(1+\dfrac{0.06}{2}\right)}{\dfrac{0.06}{2}}\\\\\\\\PV=\dfrac{700\left(1-\dfrac{1}{\left(1.03\right)^{36}}\right)\left(1.03\right)}{0.03}\\\\\\\\PV=15741.05405...\\\\\\PV=\$15,741.05\;\sf (nearest\;cent)[/tex]
Therefore, the amount that should be deposited now to yield an annuity payment of $700 at the beginning of each six months for 18 years is:
[tex]\Large\boxed{\boxed{\$15,741.05}}[/tex]
