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A young married couple has carefully looked at their budget. After review, they can afford a monthly mortgage payment of $988.00. They go to their local banker and she offers them a mortgage of 5.16% APR with monthly compounding with a term of 30 years. The couple has enough savings to pay 20% down, so the mortgage will be 80% of the home’s value. How much house can the couple afford?

3 answers

To determine how much house the couple can afford, we first need to calculate the maximum loan amount they can take out based on their monthly mortgage payment.

The mortgage formula we use to calculate the monthly payment (M) is given by:

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

Where:

  • \(M\) is the monthly payment ($988.00)
  • \(P\) is the loan amount (which we need to find)
  • \(r\) is the monthly interest rate (APR / 12)
  • \(n\) is the number of payments (loan term in months)

Step 1: Convert the interest rate and calculate \(n\)

  1. The APR is 5.16%, so the monthly interest rate \(r\) is:

\[ r = \frac{5.16%}{100} \div 12 = \frac{0.0516}{12} \approx 0.0043 \]

  1. The loan term is 30 years, so the number of payments \(n\) is:

\[ n = 30 \times 12 = 360 \]

Step 2: Rearranging the formula to find \(P\)

Now we rearrange the mortgage formula to solve for \(P\):

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

Step 3: Plug in the known values

Substituting in the known values:

\[ P = 988 \frac{(1 + 0.0043)^{360} - 1}{0.0043(1 + 0.0043)^{360}} \]

Calculating \( (1 + 0.0043)^{360} \):

\[ (1 + 0.0043)^{360} \approx 4.476 \]

Now plug in this value into the equation:

\[ P = 988 \frac{4.476 - 1}{0.0043 \times 4.476} \]

\[ P = 988 \frac{3.476}{0.019209} \]

Calculating the denominator:

\[ 0.0043 \times 4.476 \approx 0.019209 \]

Now substituting back to find \(P\):

\[ P \approx 988 \frac{3.476}{0.019209} \approx 988 \times 181.478 \approx 179,719.64 \]

Step 4: Calculate the total value of the house

Now that we have the loan amount \(P\), and knowing the couple is putting down 20%, we can calculate the total value of the house they can afford.

Let \(V\) be the total value of the house:

\[ P = 0.80V \]

So, we now solve for \(V\):

\[ V = \frac{P}{0.80} = \frac{179,719.64}{0.80} \approx 224,649.55 \]

Conclusion

The couple can afford a home worth approximately $224,649.55.

what values do I enter for N, i/y, PV, PMT, and FV to solve on a finance calculator?

To solve this problem using a financial calculator, you'll want to input the values corresponding to the parameters of the mortgage calculation. The typical inputs for a financial calculator (using PMT, PV, FV, etc.) are as follows:

  1. N (Number of Periods): This is the total number of monthly payments for the mortgage term. For a 30-year mortgage, that would be:

    • N = 360 (30 years * 12 months/year)

  2. I/Y (Interest Rate per Year): This is the annual interest rate. Since the calculator typically requires the monthly interest rate for loan payments, you should enter:

    • I/Y = 5.16 / 12 = 0.43 (or you can simply enter 5.16% and some calculators will convert it for you if you set them to monthly rates).

  3. PV (Present Value): The present value is the loan amount, which we calculated previously. Before calculating with the financial calculator, remember this is the amount they will borrow after the 20% down payment. So:

    • PV = 179,719.64 (the loan amount after down payment)

  4. PMT (Payment): This is the monthly mortgage payment, which you stated:

    • PMT = -988 (use the negative sign to indicate a cash outflow, since it’s money being paid out each month)

  5. FV (Future Value): This is the final value of the loan after all payments have been made. For a mortgage, at the end of the term (30 years), this usually is:

    • FV = 0 (as the loan is completely paid off, there should be no balance left)

Summary of Inputs:

  • N = 360
  • I/Y = 5.16 / 12 = 0.43 or 5.16% (if the calculator allows)
  • PV = 179,719.64
  • PMT = -988
  • FV = 0

With these values entered into your financial calculator, you can confirm whether your inputs and calculation align with the expected outcomes, including the present value of the loan.