Chapter 8  Home Ownership

Use TVM keys

Amortization worksheet (set calculator to 2 decimal places)

Jason Bradley’s current mortgage loan balance is $42,785. His monthly payment is $1,488, which includes taxes and insurance of $315.  How many more payments does he have on this loan?  Assume an interest rate of 6.75%.  Express your answer as a whole number of payments.

To determine how many more payments Jason Bradley has on his loan, we'll use the TVM (Time Value of Money) keys on a financial calculator. Here's how we can solve this:

1. **Extract the given dataTo determine how many more payments Jason Bradley has on his loan, we'll use the TVM (Time Value of Money) keys on a financial calculator. Here's how we can solve this:

1. **Extract the given **
   - Current mortgage loan balance ($PV$): $42,785
   - Monthly payment (excluding taxes and insurance) ($PMT$): $1,488 - $315 = $1,173
   - Annual interest rate ($I/Y$): 6.75%

2. **Convert the annual interest rate to a monthly interest rate:**
   \[
   \text{Monthly interest rate} = \frac{6.75\%}{12} = 0.5625\%
   \]

3. **Set the financial calculator to 2 decimal places.**

4. **Input the known values into the financial calculator:**
   - $PV = 42,785$
   - $PMT = -1,173$ (payments are cash outflows, so they are negative)
   - $I/Y = 6.75$ (annual interest rate)
   - $FV = 0$ (the loan will be paid off, so future value is 0)

5. **Calculate the number of payments ($N$):**
   - Use the TVM keys to compute $N$.

Let's perform the calculation:

Using the TVM formula for the number of payments:
\[
N = \frac{\ln \left(\frac{PMT}{PMT - PV \cdot \frac{I/Y}{100}}\right)}{\ln \left(1 + \frac{I/Y}{100}\right)}
\]

Where:
- $PMT = 1,173$
- $PV = 42,785$
- $I/Y = 6.75 / 12 = 0.5625$

Let's compute it step-by-step:

1. Compute the monthly interest rate in decimal form:
   \[
   \frac{0.5625}{100} = 0.005625
   \]

2. Calculate the term inside the logarithm:
   \[
   \frac{1,173}{1,173 - 42,785 \cdot 0.005625} = \frac{1,173}{1,173 - 240.918125} = \frac{1,173}{932.081875}
   \]
   \[
   \frac{1,173}{932.081875} \approx 1.2586
   \]

3. Take the natural logarithm of the term:
   \[
   \ln(1.2586) \approx 0.2293
   \]

4. Compute the denominator:
   \[
   \ln(1 + 0.005625) = \ln(1.005625) \approx 0.00561
   \]

5. Finally, calculate $N$:
   \[
   N = \frac{0.2293}{0.00561} \approx 40.88
   \]

Since the number of payments must be a whole number, we round up:
\[
N \approx 41 \text{ payments}
\]

Jason Bradley has approximately **41 more payments** left on his loan.

Would you like any details or have any questions about this calculation? 

Here are 5 questions you might ask next:
1. How can I verify my calculations using a financial calculator?
2. What impact does changing the interest rate have on the number of payments?
3. How does including taxes and insurance in the monthly payment affect the calculation?
4. Can you explain the process of calculating monthly payments for a mortgage?
5. What are some strategies to pay off a mortgage faster?

**Tip:** Always double-check your inputs when using a financial calculator to ensure accuracy, especially with interest rates and payment values.

Learn how to calculate the remaining payments on a mortgage loan using Time Value of Money (TVM) and amortization concepts. This problem involves determining the number of payments left based on Jason Bradley's mortgage details, including loan balance, monthly payment, and interest rate. Suitable for advanced high school or college students studying finance or mathematics.

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