Betty and Bob borrow $300,000 at 12% per annum compounded monthly for a 20-year (fixed rate) mortgage.

a.     What is their monthly payment?

b.    How much interest will they pay over the 20 years?

c.     After 10 years how much equity will they have in their property, i.e. how much principal will they have paid?

To solve the mortgage problem, we'll calculate the monthly payment, total interest paid, and the equity in the property after 10 years using the formula for monthly payments on a fixed-rate mortgage.

### a. Monthly Payment Calculation

The formula for the monthly payment $ M $ on a fixed-rate mortgage is:

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

where:
- $ P = 300,000 $ is the principal loan amount.
- $ r = \frac{12\%}{12} = 1\% = 0.01 $ is the monthly interest rate (since it is compounded monthly).
- $ n = 20 \times 12 = 240 $ is the total number of payments (months).

Let's calculate the monthly payment $ M $.

### b. Total Interest Paid Over 20 Years

The total interest paid over the life of the mortgage is:

\[
\text{Total Interest} = (M \times n) - P
\]

where:
- $ M $ is the monthly payment calculated in part (a).
- $ n = 240 $ is the total number of payments.
- $ P = 300,000 $ is the principal.

### c. Principal Paid After 10 Years

To find the equity after 10 years, we need to calculate the remaining balance after 10 years (120 months).

The remaining balance $ B $ after $ m $ payments can be calculated using:

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

where:
- $ m = 120 $ (10 years of payments).

The equity in the property after 10 years will be the initial principal minus the remaining balance:

\[
\text{Equity} = P - B
\]

Let's perform these calculations.### Solutions:

a. **Monthly Payment:** Betty and Bob's monthly payment will be approximately **$3,303.26**.

b. **Total Interest Paid Over 20 Years:** The total interest paid over the 20 years will be around **$492,782.02**.

c. **Equity After 10 Years:** After 10 years, the equity in their property will be approximately **$69,761.17**.

Would you like more details on the calculations, or do you have any other questions?

Here are 5 related questions that you might find interesting:
1. How would the monthly payment change if the interest rate were 10% per annum?
2. What would be the impact of making an additional payment of $100 per month?
3. How much equity would they have after 15 years instead of 10 years?
4. What is the effect of a shorter mortgage term (e.g., 15 years) on the total interest paid?
5. How does the payment schedule change if the interest is compounded semi-annually instead of monthly?

**Tip:** When considering a mortgage, always compare the total interest paid over different terms and rates to find the most cost-effective option.

Betty and Bob borrow $300,000 at 12% per annum compounded monthly for a 20-year (fixed rate) mortgage. a. What is their monthly payment? b. How much interest will they pay over the 20 years? c. After 10 years how much equity will they have in their property, i.e. how much principal will they have paid?

Calculate the monthly payment, total interest, and equity in a $300,000 fixed-rate mortgage at 12% interest compounded monthly over 20 years. Find out how much Betty and Bob will pay monthly, the interest they'll pay over the loan's life, and their equity after 10 years.

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