3.4 Mortgages

Step 1: Calculate the loan amount.

[latex]\begin{eqnarray*} \mbox{Loan Amount} & = & 408,726-50,000 \\ & = & \$358,726 \end{eqnarray*}[/latex]

Step 2: Calculate the payment.

[latex]PMT=\$2,145.98[/latex]

The monthly mortgage payments are [latex]\$2,145.98[/latex].

Step 3: Calculate the balance after five years.

To find the balance after five years, enter the payment number that corresponds to the last payment made in year five. There are [latex]12[/latex] payments a year, so the last payment made in year five is [latex]60[/latex] ([latex]5\times 12[/latex]). So, to find the balance after five years, set [latex]P_1=60[/latex] and [latex]P_2=60[/latex].

[latex]BAL=\$318,927.89[/latex]

Step 4: Write as a statement.

At the end of the five year term, the balance is [latex]\$318,927.89[/latex].

Step 1: Calculate the payment for the first term.

[latex]PMT=\$2,535.26[/latex]

The monthly mortgage payments are [latex]\$2,535.26[/latex].

Step 2: Calculate the balance at the end of the three year term.

To find the balance after three years, enter the payment number that corresponds to the last payment made in year three. There are [latex]12[/latex] payments a year, so the last payment made in year three is [latex]36[/latex] ([latex]3\times 12[/latex]). So, to find the balance after three years, set [latex]P_1=36[/latex] and [latex]P_2=36[/latex].

[latex]BAL=\$351,770.37[/latex]

At the end of the three year term, the balance is [latex]\$351,770.37[/latex].

Step 3: Calculate the payments when the mortgage is renewed.

The interest rate changes to [latex]5.85\%[/latex] compounded semi-annually. The time used to find the new payments is based on the remaining amortization period. The original amortization period is [latex]20[/latex] years and [latex]3[/latex] years have past to get to the time of renewal, so the remaining amortization period is [latex]20-3=17[/latex] years. The new payments are calculated using the new interest rate, [latex]5.85\%[/latex] compounded semi-annually, the [latex]17[/latex] years remaining on the amortization period, and the remaining balance, [latex]\$351,770.37[/latex], from the end of the previous term.

[latex]PMT=\$2,711.92[/latex]

Step 4: Write as a statement.

The monthly payments when the mortgage is renewed are [latex]\$2,711.92[/latex].

Step 1: Calculate the amortization period for the weekly payments.

[latex]N=1,211.952... \rightarrow 1,212 \mbox{ payments}[/latex]

Convert (rounded UP) [latex]N[/latex] to years and months.

[latex]\begin{eqnarray*} \mbox{Number of Years} & = & \frac{\mbox{rounded up }N}{P/Y} \\ & = & \frac{1,212}{52} \\ & = & 23.307... \\ & \rightarrow & 23 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.307... \times 12 \\ & = & 3.692... \\ & \rightarrow & 4 \mbox{ months} \end{eqnarray*}[/latex]

The amortization period for the weekly payments is [latex]23[/latex] years, [latex]4[/latex] months.

Step 2: Calculate the difference in the amortization periods.

[latex]\begin{eqnarray*} \mbox{Difference} & = & 25 \mbox{ years}-(23 \mbox{ years and } 4 \mbox{ months}) \\ & = & 1 \mbox { year and } 8 \mbox{ months} \end{eqnarray*}[/latex]

Step 3: Write as a statement.

With the weekly payments, the amortization period is shortened by [latex]1[/latex] year and [latex]8[/latex] months.

Step 1: Calculate the monthly payments.

[latex]PMT=\$1,997.08[/latex]

The monthly mortgage payments are [latex]\$1,997.08[/latex].

Step 2: Calculate the balance at the end of five years (the time of the lump-sum payment).

To find the balance after five years, enter the payment number that corresponds to the last payment made in year five. There are [latex]12[/latex] payments a year, so the last payment made in year five is [latex]60[/latex] ([latex]5\times 12[/latex]). So, to find the balance after five years, set [latex]P_1=60[/latex] and [latex]P_2=60[/latex].

[latex]BAL=\$345,120.01[/latex]

After five years, the balance is [latex]\$345,120.01[/latex].

Step 3: Subtract the lump-sum payment from the balance found in the previous step.

[latex]\begin{eqnarray*}\mbox{New Balance} & = & 345,120.01-15,000 \\ & = & \$330,120.01 \end{eqnarray*}[/latex]

Step 4: Calculate the number of payments needed to re-pay the new balance found in the previous step.

[latex]N=225.263... \rightarrow 226 \mbox{ payments}[/latex]

Step 5: Calculate the difference between the number of payments without the lump-sum payment and the number of payments with the lump-sum payment.

There are [latex]300[/latex] payments without the lump-sum payment applied. With the lump-sum payment applied, there are [latex]60[/latex] payments before the lump-sum payment and [latex]226[/latex] payment after. So with the lump-sum payment, there are a total of [latex]226+60=286[/latex] payments.

[latex]\begin{eqnarray*} \mbox{Difference} & = & 300-286 \\ & = & 14 \mbox { payments} \end{eqnarray*}[/latex]

With the lump-sum payment applied, it takes [latex]14[/latex] fewer payments to pay-off the mortgage.

Step 6: Convert the difference found in the previous step to years and months.

[latex]\begin{eqnarray*} \mbox{Number of Years} & = & \frac{14}{12} \\ & = & 1.1666... \\ & \rightarrow & 1 \mbox{ year} \\ \\ \mbox{Number of Months} & = & 0.16666... \times 12 \\ & = & 2 \mbox{ months} \end{eqnarray*}[/latex]

Step 7: Write as a statement.

The amortization period is shortened by [latex]1[/latex] year and [latex]2[/latex] months.

Step 1: Calculate the monthly payments.

[latex]PMT=\$2,094.82[/latex]

The monthly mortgage payments are [latex]\$2,094.82[/latex].

Step 2: Calculate the balance at the end of eight years (the time of the payment increase).

To find the balance after eight years, enter the payment number that corresponds to the last payment made in year eight. There are [latex]12[/latex] payments a year, so the last payment made in year eight is [latex]96[/latex] ([latex]8\times 12[/latex]). So, to find the balance after eight years, set [latex]P_1=96[/latex] and [latex]P_2=96[/latex].

[latex]BAL=\$251,916.89[/latex]

After eight years, the balance is [latex]\$251,916.89[/latex].

Step 3: Add the payment increase to the payment to find the new payment amount.

[latex]\begin{eqnarray*}\mbox{New Payment} & = & 2,094.82+200 \\ & = & \$2,294.82 \end{eqnarray*}[/latex]

Step 4: Calculate the number of payments needed to re-pay the balance found in Step 2 with the new payment found in Step 3.

[latex]N=129.108... \rightarrow 130 \mbox{ payments}[/latex]

Step 5: Calculate the difference between the number of payments without the payment increase and the number of payments with the payment increase.

There are [latex]240[/latex] payments without the payment increase. With the payment increase, there are [latex]96[/latex] payments before the payment increase and [latex]130[/latex] payment after. So with the payment increase, there are a total of [latex]96+130=226[/latex] payments.

[latex]\begin{eqnarray*} \mbox{Difference} & = & 240-226 \\ & = & 14 \mbox { payments} \end{eqnarray*}[/latex]

With the payment increase, it takes [latex]14[/latex] fewer payments to pay-off the mortgage.

Step 6: Convert the difference found in the previous step to years and months.

[latex]\begin{eqnarray*} \mbox{Number of Years} & = & \frac{14}{12} \\ & = & 1.1666... \\ & \rightarrow & 1 \mbox{ year} \\ \\ \mbox{Number of Months} & = & 0.16666... \times 12 \\ & = & 2 \mbox{ months} \end{eqnarray*}[/latex]

Step 7: Write as a statement.

The amortization period is shortened by [latex]1[/latex] year and [latex]2[/latex] months.