Question

Question 6
You have just applied for a 30-year $100,000 mortgage at a rate of 10%. What must the annual payment be?

*Make sure to input all currency answers without any currency symbols or commas, and use two decimal places of precision.

Answer 1

To calculate the annual payment for a mortgage, we can use the formula for the annuity payment, which is:

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

Where:
- \( P \) is the annual payment
- \( PV \) is the present value or principal amount of the loan (in this case, $100,000)
- \( r \) is the annual interest rate (as a decimal)
- \( n \) is the total number of payments (for a 30-year mortgage with annual payments, this will be 30)

Given:
- \( PV = 100,000 \)
- Annual interest rate \( = 10\% = 0.10 \)
- \( n = 30 \)

We substitute these values into the formula:

1. First, we calculate \( r \):
\[
r = 0.10
\]

2. Substitute into the payment formula:
\[
P = \frac{0.10 \times 100,000}{1 - (1 + 0.10)^{-30}}
\]

3. Calculate \( (1 + 0.10)^{-30} \):
\[
(1 + 0.10)^{-30} = (1.10)^{-30} \approx 0.057308
\]
Therefore:
\[
1 - (1 + 0.10)^{-30} \approx 1 - 0.057308 \approx 0.942692
\]

4. Now substitute back into the payment formula:
\[
P = \frac{0.10 \times 100,000}{0.942692} \approx \frac{10,000}{0.942692} \approx 1060.66
\]

Thus, the annual payment must be approximately:

\[
\text{Annual Payment} = 1060.66
\]

So the final answer is:

\[
1060.66
\]