To solve for the present value (PV) using the TVM Solver in a calculator, we need to identify the correct formula that matches the given parameters. The parameters provided are:
- [tex]\(N=96\)[/tex] — number of compounding periods
- [tex]\(I\%=5.4\)[/tex] — annual interest rate
- [tex]\(PMT=-560\)[/tex] — payment amount (negative because it’s an outflow)
- [tex]\(FV=0\)[/tex] — future value
- [tex]\(P/Y=12\)[/tex] — payments per year
- [tex]\(C/Y=12\)[/tex] — compounding periods per year
Firstly, we convert the annual interest rate to a monthly interest rate since the payments and compounding periods are monthly:
[tex]\[
\text{Monthly interest rate} = \frac{I\%}{P/Y} = \frac{5.4\%}{12} = 0.45\% = 0.0045
\][/tex]
The necessary formula to find the present value [tex]\(PV\)[/tex] of an annuity (given that the payments are made at the end of each period) is as follows:
[tex]\[
PV = PMT \times \frac{(1 + r)^N - 1}{r(1 + r)^N}
\][/tex]
where [tex]\(r\)[/tex] is the monthly interest rate and [tex]\(N\)[/tex] is the total number of payments.
Substituting the given values:
[tex]\[
PMT = -560, \quad r = 0.0045, \quad N = 96
\][/tex]
[tex]\[
PV = -560 \times \frac{(1 + 0.0045)^{96} - 1}{0.0045 (1 + 0.0045)^{96}}
\][/tex]
Let us match this formula to the given expressions. Among the options, we need to find the one that fits the above calculation. Focus on the correct use of [tex]\(r\)[/tex], [tex]\(N\)[/tex], PMT and check their consistency.
Expression (B) matches the formula structure with the correct values:
[tex]\[
\frac{(\$ 560) \left((1+0.0045)^{96}-1\right)}{(0.0045)(1+0.0045)^{96}}
\][/tex]
The correct choice, which will give the same value for [tex]\(PV\)[/tex], is:
[tex]\[
\boxed{\frac{(\$ 560) \left((1+0.0045)^{96}-1\right)}{(0.0045)(1+0.0045)^{96}}}
\][/tex]
