To determine the monthly payment for a 25-year loan of [tex]$175,000 at an annual interest rate of 6.7%, compounded monthly, using the TVM (Time Value of Money) Solver on a graphing calculator, we need to identify the correct parameters to input.
### Explanation
1. N (Total Number of Payments):
This is the total number of monthly payments over the term of the loan. For a 25-year loan with monthly payments, you multiply the number of years by the number of payments per year:
\[
N = 25 \text{ years} \times 12 \text{ months/year} = 300 \text{ payments}
\]
2. I% (Monthly Interest Rate):
The annual interest rate given is 6.7%, but we need the monthly interest rate. Since there are 12 months in a year, we divide the annual rate by 12:
\[
I\% = \frac{6.7\%}{12} \approx 0.558\%
\]
3. PV (Present Value):
The present value represents the amount of the loan. Here, it is given as $[/tex]175,000. Since the loan amount is money that you receive upfront (outflow from the lender’s perspective), it is negative:
[tex]\[
PV = -175,000
\][/tex]
4. FV (Future Value):
At the end of the loan period, the future value is typically zero because the loan is fully paid off:
[tex]\[
FV = 0
\][/tex]
5. PMT (Payment):
This is the unknown value we need to calculate.
6. P/Y (Payments per Year) and C/Y (Compounding periods per Year):
Since payments and compounding both occur monthly, we set:
[tex]\[
P/Y = 12, \quad C/Y = 12
\][/tex]
7. PMT Mode:
Payments are made at the end of each period, which is standard for most loans:
[tex]\[
PMT: END
\][/tex]
### Correct Setup
Given the choices, we need to match the parameters mentioned above:
- Option A: This erroneously sets PV as 0 and FV as -175,000, which would be incorrect for this context.
- Option C and Option D: Both of these options incorrectly set [tex]\( N \)[/tex] as 25, which would mean yearly payments rather than monthly.
- Option B: This accurately reflects our findings:
- [tex]\( N = 300 \)[/tex]
- [tex]\( 1 \% = 6.7 \)[/tex] (annual rate given, but we assume it's handled correctly by the calculator setup)
- [tex]\( PV = -175,000 \)[/tex]
- [tex]\( FV = 0 \)[/tex]
- [tex]\( P/Y = 12 \)[/tex]
- [tex]\( C/Y = 12 \)[/tex]
- [tex]\( PMT: END \)[/tex]
Therefore, the correct answer is:
B. [tex]\( N = 300 ; I\% = 6.7 ; PV = -175,000 ; PMT = ; FV = 0 ; P/Y = 12 ; C/Y = 12 ; PMT: END \)[/tex]
