To determine which expression calculates the same present value (PV) from the given data, let's examine each option provided.
Given data:
- [tex]\( N = 96 \)[/tex] (total number of payments)
- Annual interest rate: [tex]\( 5.4\% \)[/tex] converted to a decimal [tex]\( = 0.054 \)[/tex]
- [tex]\( PMT = -560 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- [tex]\( P / Y = 12 \)[/tex] (payments per year)
- [tex]\( C / Y = 12 \)[/tex] (compounding periods per year)
- Payments occur at the end of the period: PMT:END.
1. Convert the annual interest rate to a monthly interest rate:
[tex]\[ \text{Monthly interest rate} = \frac{\text{Annual rate}}{P / Y} = \frac{0.054}{12} = 0.0045 \][/tex]
2. Confirm the number of payments:
[tex]\[ N = 8 \times 12 = 96 \][/tex] (as already given)
Next, let's evaluate each expression with the given data:
### Option A:
[tex]\[ \frac{560\left((1+0.054)^{0}-1\right)}{(0.054)(1+0.054)^{0}} \][/tex]
[tex]\[
= \frac{560 \left(1 - 1\right)}{0.054 \cdot 1}
= \frac{560 \cdot 0}{0.054}
= 0
\][/tex]
### Option B:
[tex]\[ \frac{560\left((1+0.0045)^{96}-1\right)}{(0.0045)(1+0.0045)^{96}} \][/tex]
[tex]\[
= \frac{560 \left((1+0.0045)^{96} - 1\right)}{0.0045 \cdot (1+0.0045)^{96}}
\][/tex]
The calculation of this expression will result in a negative value which is approximately:
[tex]\[ \approx -43,575.61 \][/tex]
### Option C:
[tex]\[ \frac{560\left((1+0.054)^0 -1\right)}{(0.054)(1+0.054)^6} \][/tex]
[tex]\[
= \frac{560 \left(1 - 1\right)}{(0.054)(1+0.054)^6}
= \frac{560 \cdot 0}{0.054 \cdot (1.3561)}
= 0
\][/tex]
### Option D:
[tex]\[ \frac{560\left((1+0.0045)^8 -1\right)}{(0.0045)(1+0.0045)^8} \][/tex]
[tex]\[
= \frac{560 \left( (1+0.0045)^8 -1 \right)}{(0.0045)(1+0.0045)^8}
\][/tex]
The calculation of this expression will result in a negative value which is approximately:
[tex]\[ \approx -4,390.62 \][/tex]
### Conclusion:
The expression that returns a value for PV that matches is Option B.
[tex]\[ \frac{560\left((1+0.0045)^{96}-1\right)}{(0.0045)(1+0.0045)^{96}} \approx -43,575.61 \][/tex]
Option B is the correct answer.
