Determine the maximum​ 30-year fixed-rate mortgage amount for which a couple could qualify if the rate is 4.58 percent. Assume they have other debt payments totaling ​$417 per month and a combined annual income of ​$55 comma 900. Monthly escrow payments for real estate taxes and​ homeowner's insurance are estimated to be ​$226. ​(Assume a 36 percent maximum of annual income for total debt and escrow​ payments.)

To determine the maximum 30-year fixed-rate mortgage amount the couple can qualify for, we need to calculate their allowable monthly debt payment (including the mortgage, other debts, and escrow payments), then use that amount to calculate the maximum mortgage payment.

### Steps:

1. **Determine the total monthly debt limit:**
   The couple's total debt payments (including the mortgage payment, other debts, and escrow) should not exceed 36% of their combined annual income.

   \[
   \text{Maximum debt-to-income ratio} = 36\% \times \text{Annual income}
   \]
   Given the annual income of $55,900:
   \[
   \text{Maximum debt-to-income ratio} = 0.36 \times 55,900 = 20,124
   \]
   This is the maximum allowable annual total debt and escrow payment.

   Now, convert this to a monthly limit:
   \[
   \frac{20,124}{12} = 1,677
   \]
   So, their total monthly debt payment (including mortgage, other debt, and escrow) must not exceed $1,677.

2. **Subtract other monthly obligations:**
   - Other debt payments = $417 per month.
   - Escrow payments (taxes and insurance) = $226 per month.

   Total non-mortgage payments = $417 + $226 = $643 per month.

3. **Calculate the maximum monthly mortgage payment:**
   The mortgage payment is the remaining amount after subtracting other debts and escrow from the total monthly debt limit.
   \[
   \text{Maximum mortgage payment} = 1,677 - 643 = 1,034
   \]

4. **Calculate the maximum mortgage amount:**
   Now, we'll use the maximum monthly mortgage payment ($1,034) to find the maximum mortgage loan amount. The formula for the monthly mortgage payment on a fixed-rate loan is:

   \[
   M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
   \]
   Where:
   - $ M $ is the monthly mortgage payment ($1,034),
   - $ P $ is the loan amount (what we want to find),
   - $ r $ is the monthly interest rate (annual rate of 4.58% divided by 12),
   - $ n $ is the number of monthly payments (30 years $\times$ 12 = 360 months).

   First, calculate the monthly interest rate:
   \[
   r = \frac{4.58\%}{12} = 0.00381667
   \]

   Now use the mortgage payment formula. We need to solve for $ P $, which gives the maximum mortgage amount:

   \[
   1,034 = P \times \frac{0.00381667(1 + 0.00381667)^{360}}{(1 + 0.00381667)^{360} - 1}
   \]

   The factor $ \frac{0.00381667(1 + 0.00381667)^{360}}{(1 + 0.00381667)^{360} - 1} $ simplifies to approximately 0.005119.

   So:
   \[
   1,034 = P \times 0.005119
   \]
   Solving for $ P $:
   \[
   P = \frac{1,034}{0.005119} \approx 202,000
   \]

### Conclusion:
The couple can qualify for a maximum 30-year fixed-rate mortgage of approximately **$202,000**.

Would you like a more detailed breakdown or clarification on any step?

### Related Questions:
1. How does the interest rate affect the mortgage amount they can qualify for?
2. What would happen if their combined annual income were higher or lower?
3. How would a change in their debt payments impact the mortgage qualification?
4. What if the mortgage rate was 5% instead of 4.58%?
5. How does the length of the mortgage (e.g., 15 vs. 30 years) affect monthly payments?

### Tip:
Always consider the impact of property taxes and insurance when calculating how much house you can afford, as they significantly impact your monthly budget.

Determine the maximum 30-year fixed-rate mortgage amount for which a couple could qualify if the rate is 4.58 percent. Assume they have other debt payments totaling $417 per month and a combined annual income of $55,900. Monthly escrow payments for real estate taxes and homeowner's insurance are estimated to be $226. (Assume a 36 percent maximum of annual income for total debt and escrow payments.)

Learn how to calculate the maximum 30-year fixed-rate mortgage a couple can afford with a 4.58% interest rate, a combined annual income of $55,900, and monthly debt payments. This example covers key concepts such as debt-to-income ratios and loan amortization, ideal for advanced students or financial planning enthusiasts.

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