- The monthly mortgage payment \(M\) is calculated using the formula \(M = P \frac{r(1+r)^n}{(1+r)^n - 1}\).
- Statement I is true because higher interest rates increase the monthly payment factor.
- Statement II is false because a higher down payment reduces the loan principal, which decreases the monthly payment.
- Statement III is true because a longer term spreads the loan repayment over more months, resulting in smaller individual payments.
- The correct option is \(\boxed{D}\).
Explanation
- Analyzing the Mortgage Payment Formula
The standard monthly mortgage payment formula is given by:
\[M = P \frac{r(1+r)^n}{(1+r)^n - 1}\]
where:
- \(M\) is the monthly mortgage payment.
- \(P\) is the principal loan amount, which is equal to the home price minus the down payment.
- \(r\) is the monthly interest rate (annual interest rate divided by 12).
- \(n\) is the total number of payments in months.
We need to analyze three statements based on this formula.
- Analysis of Statement I
Statement I: The higher your interest rate, the higher your monthly mortgage payments.
In the formula, as the interest rate \(r\) increases, both the numerator and the denominator change. Specifically, the factor \(\frac{r(1+r)^n}{(1+r)^n - 1}\) is an increasing function of \(r\). Since the principal \(P\) and the number of payments \(n\) are held constant, an increase in the interest rate \(r\) leads to an increase in the monthly payment \(M\). Therefore, this statement is true.
- Analysis of Statement II
Statement II: The higher your down payment, the higher your monthly mortgage payments.
The principal loan amount \(P\) is defined as the home price minus the down payment:
\[P = \text{Home Price} - \text{Down Payment}\]
If the down payment increases, the principal \(P\) must decrease. Since the monthly payment \(M\) is directly proportional to the principal \(P\) (assuming \(r\) and \(n\) are constant), a decrease in \(P\) results in a decrease in the monthly payment \(M\). Thus, a higher down payment leads to lower monthly payments, not higher ones. This statement is false.
- Analysis of Statement III
Statement III: A 30-year mortgage fixed at 6% will have smaller payments than a 20-year mortgage fixed at 6%.
Both mortgages have the same interest rate \(r\) and the same principal \(P\). The only difference is the duration \(n\). A 30-year mortgage has \(n = 30 \times 12 = 360\) months, while a 20-year mortgage has \(n = 20 \times 12 = 240\) months.
Let \(f(n) = \frac{r(1+r)^n}{(1+r)^n - 1} = \frac{r}{1 - (1+r)^{-n}}\).
As \(n\) increases, \((1+r)^{-n}\) decreases, which means \(1 - (1+r)^{-n}\) increases. Consequently, \(f(n)\) decreases as \(n\) increases. Therefore, the payment factor for 360 months is smaller than the payment factor for 240 months, resulting in smaller monthly payments for the 30-year loan. This statement is true.
- Conclusion
We have established that:
- Statement I is true.
- Statement II is false.
- Statement III is true.
Thus, statements I and III are true. This corresponds to choice (d). Consequently, the correct option is D.
Examples
When planning to buy a home, understanding how your down payment and loan term affect your monthly budget is crucial. For example, if you are struggling to make monthly payments, you might choose to increase your down payment (which lowers the loan principal) or extend your mortgage term (e.g., from 20 to 30 years). Using the mortgage formula \(\$M = P \frac{r(1+r)^n}{(1+r)^n - 1}\$\), you can calculate exactly how these changes impact your finances and ensure your mortgage is affordable in the long term.