Five years ago, Diane secured a bank loan of $390,000 to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was 30 years, and the interest rate was 9%/year compounded monthly on the unpaid balance. Because the interest rate for a conventional 30-year home mortgage has now dropped to 5.5%/year compounded monthly, Diane is thinking of refinancing her property. (Round your answers to the nearest cent.)
(a) What is Diane's current monthly mortgage payment?
$

(b) What is Diane's current outstanding principal?
$

(c) If Diane decides to refinance her property by securing a 30-year home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of 5.5%/year compounded monthly, what will be her monthly mortgage payment?
$

(d) How much less will Diane's monthly mortgage payment be if she refinances?

To solve this, we'll break down the problem into steps. Diane's current mortgage is a standard fixed-rate mortgage with monthly compounding and payments. Let's address each part systematically.

### (a) Current Monthly Mortgage Payment

The formula to calculate monthly payments on a mortgage is:

\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]

Where:
- $M$ = Monthly payment
- $P$ = Loan amount (principal), which is $390,000
- $r$ = Monthly interest rate, which is the annual rate divided by 12
- $n$ = Total number of payments (30 years $\times$ 12 months = 360)

For Diane's original loan:
- $P = 390,000$
- Annual interest rate = 9% → Monthly rate $r = \frac{9\%}{12} = \frac{0.09}{12} = 0.0075$
- $n = 30 \times 12 = 360$

Using this, we can calculate her current monthly payment.

### (b) Current Outstanding Principal

To find the current outstanding principal after 5 years, we use the amortization formula, which takes into account how much of the loan has been paid off. After 5 years, Diane has made $60$ payments, so $300$ payments remain.

The formula for the remaining balance $B$ after $k$ payments is:

\[
B = P \times \frac{(1 + r)^n - (1 + r)^k}{(1 + r)^n - 1}
\]

Where:
- $k$ is the number of payments already made (5 years $\times$ 12 = 60)

### (c) New Monthly Mortgage Payment After Refinancing

If Diane refinances the remaining principal at a lower interest rate (5.5%), we recalculate her new monthly payment using the same monthly payment formula but with the new interest rate and the remaining principal.

New conditions for the refinanced loan:
- New annual interest rate = 5.5% → Monthly rate = $\frac{5.5\%}{12} = \frac{0.055}{12} = 0.0045833$
- Loan term remains 30 years (360 months)

### (d) Monthly Savings from Refinancing

The savings in monthly payments will be the difference between the current monthly mortgage payment and the new monthly mortgage payment after refinancing.

Now, let's calculate all these values.Here are the results:

### (a) Diane's current monthly mortgage payment:
\[
\text{Current Monthly Payment} = \$3,138.03
\]

### (b) Diane's current outstanding principal after 5 years:
\[
\text{Outstanding Principal} = \$373,932.53
\]

### (c) If Diane refinances her property, her new monthly mortgage payment will be:
\[
\text{New Monthly Payment} = \$2,123.15
\]

### (d) Diane's monthly savings if she refinances:
\[
\text{Monthly Savings} = \$1,014.88
\]

Would you like more details or have any questions? Here are some related questions:

1. What are the advantages of refinancing a mortgage?
2. How does the amortization schedule affect monthly payments?
3. Can you calculate the total interest paid before and after refinancing?
4. What are the potential risks of refinancing a home loan?
5. How does the duration of a loan affect total interest paid?

**Tip:** Refinancing can lower your payments, but consider upfront costs like closing fees when calculating overall savings.

Five years ago, Diane secured a bank loan of $390,000 to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was 30 years, and the interest rate was 9%/year compounded monthly on the unpaid balance. Because the interest rate for a conventional 30-year home mortgage has now dropped to 5.5%/year compounded monthly, Diane is thinking of refinancing her property. (Round your answers to the nearest cent.) (a) What is Diane's current monthly mortgage payment? (b) What is Diane's current outstanding principal? (c) If Diane decides to refinance her property by securing a 30-year home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of 5.5%/year compounded monthly, what will be her monthly mortgage payment? (d) How much less will Diane's monthly mortgage payment be if she refinances?

Learn how to calculate Diane's monthly mortgage payments, outstanding principal after 5 years, and potential savings through refinancing. This problem covers key concepts in loan amortization, interest compounding, and mortgage refinancing. Perfect for college-level finance or algebra students.

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