# 3.4 Mortgages

## Learning Objectives

- Calculate mortgage payments for the initial term and renewal term
- Calculate amortization periods to reflect changes in payment frequency
- Calculate mortgage balances and amortization periods to reflect prepayments of principal

**Formula & Symbol Hub**

#### Symbols Used

- [latex]BAL[/latex] = Principal balance immediately after a loan payment
- [latex]INT[/latex] = Interest portion of a loan payment or a series of payments
- [latex]PRN[/latex] = Principal portion of a loan payment or a series of payments
- [latex]PMT[/latex] = Annuity payment amount
- [latex]I/Y[/latex] = Nominal interest rate
- [latex]P/Y[/latex] = Number of payments per year or payment frequency
- [latex]C/Y[/latex] = Number of compounds per year or compounding frequency
- [latex]N[/latex] = Total number of annuity payments

**Introduction**

A **mortgage** is a special type of loan that is collaterally secured by real estate. In essence, the loan has a lien against the property, which is the right to seize the property for the debt to be satisfied. An individual or business taking out a mortgage is obliged to pay back the amount of the loan with interest based on a predetermined contract. The lender, a bank, or some other financial institution, has a claim on the real estate property in the event that the mortgage goes into default, which means that it is not paid as per the agreement. In these instances, financial institutions will pursue foreclosure of the property, which allows for the tenants to be evicted and the property to be sold. The proceeds of the sale are then used to pay off the mortgage. A mortgage always involves two parties. The individual or business that borrows the money is referred to as the **mortgagor**, and the financial institution that lends the money is referred to as the **mortgagee**.

## Types of Mortgages

In the mortgage contract, the two parties can agree to either a fixed interest rate or a variable interest rate. Either way, the mortgage always forms an ordinary annuity because interest is not payable in advance.

- Under a
, the principal is repaid through a number of equal payments that cover both the interest and principal components of the loan. The interest portion is highest at the beginning and gradually declines over the amortization period of the mortgage. In Canada, fixed interest rates are either annually or semi-annually compounded, with the latter being the prevailing choice.**fixed interest rate** - In a
mortgage, the principal is repaid through an agreed-upon number of unequal payments that fluctuate with changes in borrowing rates. The principal and interest portions of the payment vary as interest rates fluctuate, meaning that the interest portion can rise at any point with any increase in rates. When rates change, a common practice in many financial institutions is to change the variable interest rate as of the first day of the next month. If the rate change does not coincide with a mortgage payment date, then the interest portion is calculated in a way similar to the procedure for a demand loan, where the exact number of days at the different rates must be determined. In Canada, variable interest rates are usually compounded monthly.**variable interest rate**

The mortgage agreement can be open or closed. An **open mortgage** has very few rules and it allows the mortgagor to pay off the debt in full or make additional prepayments at any given point in any amount without penalty. A **closed mortgage** has many rules that determine how the mortgage is to be paid. It does not allow the mortgager to pay off the debt in full until the loan matures. As a marketing tool, most closed mortgages have “top-up” options that allow the mortgagor to make additional payments (such as an additional [latex]20\%[/latex] per year) against the mortgage without penalty. Any payments exceeding the maximums or early payment of the mortgage are penalized heavily, with a three-month minimum interest charge that could be increased up to a measure called the interest rate differential, which effectively assesses the bank’s loss and charges the mortgagor this full amount.

## Calculating Mortgage Payments and Outstanding Balance

Previously, we learned how to calculate the payment for a loan and how to find the remaining balance on a loan after any payment is made. Because a mortgage is a special type of loan, we can use these same techniques on a mortgage.

### Example 3.4.1

A new home is purchased for [latex]\$408,726[/latex] with a [latex]\$50,000[/latex] down payment and a 25-year mortgage for the outstanding balance. The mortgage is repaid with monthly payments. The interest rate on the mortgage is fixed at [latex]5.29\%[/latex] compounded semi-annually for a five year term.

- What is the size of the monthly payments?
- What is the balance on the mortgage at the end of the five year term?

**Solution**

*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.**

PMT Setting |
END |

[latex]N[/latex] |
[latex]12 \times 25=300[/latex] |

[latex]PV[/latex] |
[latex]358,726[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]5.29[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[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].

PMT Setting |
END |

[latex]N[/latex] |
[latex]300[/latex] |

[latex]PV[/latex] |
[latex]358,726[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-2,145.98[/latex] |

[latex]I/Y[/latex] |
[latex]5.29[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]P_1[/latex] |
[latex]60[/latex] |

[latex]P_2[/latex] |
[latex]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].

**Key Takeaways**

## Renewing the Mortgage

When the term of a mortgage expires, the balance remaining becomes due in full. Typically the balance owing is still quite substantial, so the mortgage must be renewed. As discussed earlier, this means that the mortgagor assumes another mortgage, not necessarily with the same financial institution, and the amortization term is typically reduced by the length of the first term. The length of the second term of the mortgage then depends on the choice of the mortgagor. Other variables such as payment frequency and the interest rate may or may not change.

For example, assume a mortgage is initially taken out with a 25-year amortization period and a five-year term. After five years, the mortgage becomes due in full. Unable to pay it, the mortgagor renews the mortgage for the remaining 20-year amortization period, and also opts for a three-year term in assuming the new mortgage. When those three years are over, the mortgagor renews the mortgage for the remaining 17-year amortization period and again makes another term decision. This process repeats until the debt is ultimately paid off.

### HOW TO

#### Calculate the New Payment When a Mortgage is Renewed

When the mortgage is renewed for another term, the interest rate or payment frequency may changed. Consequently, the payments on the mortgage will also change when the mortgage is renewed and the new term goes into affect. To find the new payment when the mortgage is renewed, calculate the payment using the remaining balance at the end of the previous term and the remaining time from the amortization period.

The steps to find the new payment upon renewal are:

**Step 1:** Calculate the original payment for the mortgage before the mortgage is renewed.

**Step 2:** Calculate the remaining balance at the end of the previous term.

**Step 3:** Calculate the new payment for the mortgage using the balance from the previous step as the loan amount (the present value), the remaining time on the amortization period, and incorporating any changes that occur in the new term, such as a new interest rate or change in payment frequency.

### Example 3.4.2

A [latex]\$389,000[/latex] mortgage is repaid with monthly payments and amortized over [latex]20[/latex] years. The interest rate for the first three year term is [latex]4.9\%[/latex] compounded semi-annually. At the end of the three year term the mortgage is renewed at [latex]5.85\%[/latex] compounded semi-annually. What is the size of the monthly payment when the mortgage is renewed?

**Solution**

*Step 1:***Calculate the payment for the first term.**

PMT Setting |
END |

[latex]N[/latex] |
[latex]12 \times 20=240[/latex] |

[latex]PV[/latex] |
[latex]389,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]4.9[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[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].

PMT Setting |
END |

[latex]N[/latex] |
[latex]240[/latex] |

[latex]PV[/latex] |
[latex]389,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-2,535.26[/latex] |

[latex]I/Y[/latex] |
[latex]4.9[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]P_1[/latex] |
[latex]36[/latex] |

[latex]P_2[/latex] |
[latex]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.

PMT Setting |
END |

[latex]N[/latex] |
[latex]12 \times 17=204[/latex] |

[latex]PV[/latex] |
[latex]351,770.37[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]5.85[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[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].

### Try It

1) A [latex]\$628,200[/latex] house is purchased for [latex]\$100,000[/latex] down and monthly payments for [latex]30[/latex] years. The interest rate for the first four year term is [latex]6.49\%[/latex] compounded semi-annually.

- Calculate the monthly payments.
- What is the balance on the mortgage at the end of the four year term?
- If the mortgage is renewed at [latex]6.19\%[/latex] compounded semi-annually, what is the size of the new monthly payments?

**Solution**

1. Calculate the monthly payments.

PMT Setting |
END |

[latex]N[/latex] |
[latex]12 \times 30=360[/latex] |

[latex]PV[/latex] |
[latex]528,200[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]6.49[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]PMT=\$3,305.29[/latex]

2. Calculate the balance at the end of the four year term.

PMT Setting |
END |

[latex]N[/latex] |
[latex]360[/latex] |

[latex]PV[/latex] |
[latex]528,200[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-3,305.29[/latex] |

[latex]I/Y[/latex] |
[latex]6.49[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]P_1[/latex] |
[latex]48[/latex] |

[latex]P_2[/latex] |
[latex]48[/latex] |

[latex]BAL=\$501,665.54[/latex]

3. Calculate the new payment upon renewal.

PMT Setting |
END |

[latex]N[/latex] |
[latex]12 \times 26=312[/latex] |

[latex]PV[/latex] |
[latex]501,665.54[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]6.19[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]PMT=\$3,123.63[/latex]

## Changing the Payment Frequency

Although monthly mortgage payments are the most common payment frequency, mortgages can be paid off with semi-monthly, bi-weekly, or weekly payments. By making more frequency payments, the mortgagor can reduce the amortization period and pay-off the mortgage faster.

To determine how much the amortization period is shortened by changing the payment frequency, calculate the amortization period for the new payment frequency and then calculate the difference in the two amortization periods.

### Example 3.4.3

A [latex]\$280,000[/latex] mortgage at [latex]5.3\%[/latex] compounded semi-annually is repaid with monthly payments of [latex]\$1678[/latex] amortized over [latex]25[/latex] years. How much shorter is the amortization period if weekly payments of [latex]\$400[/latex] are made instead of the monthly payments?

**Solution**

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

PMT Setting |
END |

[latex]N[/latex] |
[latex]?[/latex] |

[latex]PV[/latex] |
[latex]280,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-400[/latex] |

[latex]I/Y[/latex] |
[latex]5.3[/latex] |

[latex]P/Y[/latex] |
[latex]52[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[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.

### Try It

A [latex]\$300,000[/latex] mortgage at [latex]4.7\%[/latex] compounded semi-annually is repaid with monthly payments of [latex]\$1,925[/latex] for [latex]20[/latex] years. How much shorter is the amortization period if bi-weekly payments of [latex]\$1,000[/latex] are made instead of the monthly payments?

**Solution**

1. Calculate the amortization period for the bi-weekly payments.

PMT Setting |
END |

[latex]N[/latex] |
[latex]?[/latex] |

[latex]PV[/latex] |
[latex]300,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-1,000[/latex] |

[latex]I/Y[/latex] |
[latex]4.7[/latex] |

[latex]P/Y[/latex] |
[latex]26[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]N=430.370... \rightarrow 431 \mbox{ payments}[/latex]

[latex]\begin{eqnarray*} \mbox{Number of Years} & = & \frac{\mbox{rounded up }N}{P/Y} \\ & = & \frac{431}{26} \\ & = & 16.576... \\ & \rightarrow & 16 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.5769... \times 12 \\ & = & 6.923... \\ & \rightarrow & 7 \mbox{ months} \end{eqnarray*}[/latex]

2. Calculate the difference in the amortization periods.

[latex]\begin{eqnarray*} \mbox{Difference} & = & 20 \mbox{ years}-(16 \mbox{ years and } 7 \mbox{ months}) \\ & = & 3 \mbox { year and } 5 \mbox{ months} \end{eqnarray*}[/latex]

With the bi-weekly payments, the amortization period is shortened by [latex]3[/latex] year and [latex]5[/latex] months.

## Lump-Sum Payments

As noted above, closed mortgages allow the mortgagor to make extra payments, in addition to the regular periodic payments. Any extra payments applied to the mortgage are applied directly to the principal because the interest is paid by the regular periodic payment. Consequently, these extra payments allow the mortgagor to repay the mortgage sooner. What affect do these extra payments have on the amortization period? How much is the amortization period shortened by when extra payments are made?

One of these pre-payment options is to make a lump-sum payment. A lump-sum payment is simply a single amount of money applied to the mortgage. For example, you might make a [latex]\$10,000[/latex] lump-sum payment to your mortgage at the end of the sixth year. This means that at the end of the sixth year of the amortization period, you pay [latex]\$10,000[/latex] to the mortgage principal. What affect does this have on the amount of time it takes to re-pay the mortgage?

### HOW TO

#### Apply a Lump Sum Payment

To calculate how much the amortization period is shortened by when a lump-sum payment is applied:

**Step 1:** Find any missing information about the mortgage. For example, you might have to calculate out the payment or the number of payments, depending on what information is missing.

**Step 2:** Calculate the balance on the mortgage at the time the lump-sum payment is made.

**Step 3:** Subtract the lump-sum payment from the balance found in Step 2. This amount becomes the remaining balance on the mortgage.

**Step 4:** Calculate the number of regular periodic payments required to pay-off the balance found in step [latex]3[/latex].

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

**Step 6:** Convert the difference found in step [latex]5[/latex] to years and months to find how much the amortization period is shortened by with the lump-sum payment included.

### Things to Watch Out For

The lump sum payment is NOT counted as one of the payments in step [latex]4[/latex]. The number of payments refers to the number of regularly scheduled periodic payments.

### Example 3.4.4

A [latex]\$400,000[/latex] mortgage at [latex]3.5\%[/latex] compounded semi-annually is repaid with monthly payments for [latex]25[/latex] years. How much is the amortization period shortened by if a [latex]\$15,000[/latex] lump-sum payment is made at the end of year five?

**Solution**

**Step 1:****Calculate the monthly payments.**

PMT Setting |
END |

[latex]N[/latex] |
[latex]12 \times 25=300[/latex] |

[latex]PV[/latex] |
[latex]400,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]3.5[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[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].

PMT Setting |
END |

[latex]N[/latex] |
[latex]300[/latex] |

[latex]PV[/latex] |
[latex]400,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-1,997.08[/latex] |

[latex]I/Y[/latex] |
[latex]3.5[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]P_1[/latex] |
[latex]60[/latex] |

[latex]P_2[/latex] |
[latex]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.**

PMT Setting |
END |

[latex]N[/latex] |
[latex]?[/latex] |

[latex]PV[/latex] |
[latex]330,120.01[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-1,997.08[/latex] |

[latex]I/Y[/latex] |
[latex]3.5[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[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.

### Try It

2) A [latex]\$300,000[/latex] mortgage at [latex]2.8\%[/latex] compounded semi-annually is repaid with monthly payments for [latex]20[/latex] years. How much is the amortization period shortened by if a [latex]\$10,000[/latex] lump-sum payment is made at the end of year four?

**Solution**

1. Calculate the monthly payments.

PMT Setting |
END |

[latex]N[/latex] |
[latex]240[/latex] |

[latex]PV[/latex] |
[latex]300,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]2.8[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]PMT=\$1,631.51[/latex]

2. Calculate the balance at the end of four years.

PMT Setting |
END |

[latex]N[/latex] |
[latex]240[/latex] |

[latex]PV[/latex] |
[latex]300,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-1,631.51[/latex] |

[latex]I/Y[/latex] |
[latex]2.8[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

P[latex]1[/latex] |
[latex]48[/latex] |

P[latex]2[/latex] |
[latex]48[/latex] |

[latex]BAL=\$252,555.63[/latex]

3. Subtract the lump-sum payment from the balance.

[latex]\begin{eqnarray*}\mbox{New Balance} & = & 252,555.63-10,000 \\ & = & \$242,555.63 \end{eqnarray*}[/latex]

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

PMT Setting |
END |

[latex]N[/latex] |
? |

[latex]PV[/latex] |
[latex]242,555.63[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-1,631.51[/latex] |

[latex]I/Y[/latex] |
[latex]2.8[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]N=182.530... \rightarrow 183 \mbox{ payments}[/latex]

5. Calculate the difference between the number of payments and convert the difference to years and months.

[latex]\begin{eqnarray*} \mbox{Difference} & = & 240-(48+183) \\ & = & 9 \mbox { payments} \end{eqnarray*}[/latex]

[latex]\begin{eqnarray*} \mbox{Number of Years} & = & \frac{9}{12} \\ & = & 0.75 \\ & \rightarrow & 0 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.75 \times 12 \\ & = & 9 \mbox{ months} \end{eqnarray*}[/latex]

The amortization period is shortened by [latex]9[/latex] months.

## Increasing the Amount of the Periodic Payment

Another one of the pre-payment options is to increase the periodic payment amount. For example, you might decide to increase your periodic payment by [latex]\$10[/latex] at the end of the sixth year. This means that starting with the first payment in year seven, each of the periodic payments will be [latex]\$10[/latex] higher. Because the interest was already paid by the original periodic payment amount, this extra [latex]\$10[/latex] gest applied to the mortgage principal. What affect does this have on the amount of time it takes to re-pay the mortgage?

### HOW TO

#### Increasing the Periodic Payment

To calculate how much the amortization period is shortened by when the amount of the periodic payment is increased:

**Step 1:** Find any missing information about the mortgage. For example, you might have to calculate out the payment or the number of payments, depending on what information is missing.

**Step 2:** Calculate the balance on the mortgage at the time the payment is increased.

**Step 3:** Calculate the new payment by adding the payment increase to the original payment amount.

**Step 4:** Calculate the number of new periodic payments required to pay-off the balance found in step [latex]2[/latex].

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

**Step 6:** Convert the difference found in step [latex]5[/latex] to years and months to find how much the amortization period is shortened by with the payment increase included.

### Example 3.4.5

A [latex]\$375,000[/latex] mortgage at [latex]3.1\%[/latex] compounded semi-annually is repaid with monthly payments for [latex]20[/latex] years. How much is the amortization period shortened by if the payments are increased by [latex]\$200[/latex] at the end of year eight?

**Solution**

*Step 1:***Calculate the monthly payments.**

PMT Setting |
END |

[latex]N[/latex] |
[latex]12 \times 20=240[/latex] |

[latex]PV[/latex] |
[latex]375,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]3.1[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[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].

PMT Setting |
END |

[latex]N[/latex] |
[latex]240[/latex] |

[latex]PV[/latex] |
[latex]375,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-2,094.82[/latex] |

[latex]I/Y[/latex] |
[latex]3.1[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]P_1[/latex] |
[latex]96[/latex] |

[latex]P_2[/latex] |
[latex]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.**

PMT Setting |
END |

[latex]N[/latex] |
[latex]?[/latex] |

[latex]PV[/latex] |
[latex]251,916.89[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-2,294.82[/latex] |

[latex]I/Y[/latex] |
[latex]3.1[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[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.

### Try It

3) A [latex]\$500,000[/latex] mortgage at [latex]4.3\%[/latex] compounded semi-annually is repaid with monthly payments for [latex]25[/latex] years. How much is the amortization period shortened by if the payments are increased by [latex]\$75[/latex] at the end of year three?

**Solution**

1. Calculate the monthly payments.

PMT Setting |
END |

[latex]N[/latex] |
[latex]300[/latex] |

[latex]PV[/latex] |
[latex]500,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]?[/latex] |

[latex]I/Y[/latex] |
[latex]4.3[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]PMT=\$2,712.04[/latex]

2. Calculate the balance at the end of three years.

PMT Setting |
END |

[latex]N[/latex] |
[latex]300[/latex] |

[latex]PV[/latex] |
[latex]500,000[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-2,712.04[/latex] |

[latex]I/Y[/latex] |
[latex]4.3[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]P_1[/latex] |
[latex]36[/latex] |

[latex]P_2[/latex] |
[latex]36[/latex] |

[latex]BAL=\$464,114.56[/latex]

3. Calculate the new payment.

[latex]\begin{eqnarray*}\mbox{New Payment} & = & 2,712.04+75 \\ & = & \$2,787.04 \end{eqnarray*}[/latex]

4. Calculate the number of payments needed to re-pay the balance found in Step 2.

PMT Setting |
END |

[latex]N[/latex] |
[latex]?[/latex] |

[latex]PV[/latex] |
[latex]464,114.56[/latex] |

[latex]FV[/latex] |
[latex]0[/latex] |

[latex]PMT[/latex] |
[latex]-2,787.04[/latex] |

[latex]I/Y[/latex] |
[latex]4.3[/latex] |

[latex]P/Y[/latex] |
[latex]12[/latex] |

[latex]C/Y[/latex] |
[latex]2[/latex] |

[latex]N=252.4769... \rightarrow 253 \mbox{ payments}[/latex]

5. Calculate the difference between the number of payments and convert the difference to years and months.

[latex]\begin{eqnarray*} \mbox{Difference} & = & 300-(36+253) \\ & = & 11 \mbox { payments} \end{eqnarray*}[/latex]

[latex]\begin{eqnarray*} \mbox{Number of Years} & = & \frac{11}{12} \\ & = & 0.9166...\\ & \rightarrow & 0 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.9166... \times 12 \\ & = & 11 \mbox{ months} \end{eqnarray*}[/latex]

The amortization period is shortened by [latex]11[/latex] months.

**Section 3.4 Exercises**

- A [latex]\$434,693[/latex] mortgage is repaid with monthly payments over [latex]30[/latex] years. The interest rate for the first three year term is [latex]4.5\%[/latex] compounded semi-annually. If the mortgage is renewed at [latex]5.25\%[/latex] compounded semi-annually, what is the size of the monthly payments upon renewal?

**Solution**[latex]\$2,370.32[/latex]

- A [latex]\$318,222[/latex] mortgage is repaid with bi-weekly payments over [latex]25[/latex] years. The interest rate for the first two year term is [latex]3\%[/latex] compounded semi-annually. If the mortgage is renewed at [latex]6.8\%[/latex] compounded semi-annually, what is the size of the bi-weekly payments upon renewal?

**Solution**[latex]\$985.74[/latex]

- Five years ago, Asia purchased her [latex]\$322,000[/latex] home in Edmonton with a [latex]25-[/latex]year amortization. In her first five-year term, she made monthly payments and was charged [latex]4.89\%[/latex] compounded semi-annually. She will renew the mortgage on the same amortization timeline for a three year term at [latex]5.49\%[/latex] compounded semi-annually with monthly payments.
- Calculate the balance remaining after the first term.
- What is the new mortgage payment amount for the second term?
- What is the balance at the end of the second term?

**Solution**a. [latex]\$284,498.75[/latex]; b. [latex]\$1,945.52[/latex]; c. [latex]\$258,813.11[/latex]

- Luke took out a [latex]\$350,000[/latex] mortgage at [latex]3.7\%[/latex] compounded semi-annually and repaid the mortgage with monthly payments of [latex]\$1,700[/latex].
- How long will it take Luke to repay the mortgage?
- How much would the amortization period be shortened by if Luke made weekly payments of [latex]\$425[/latex] instead of the monthly payments?

**Solution**a. [latex]27[/latex] years, [latex]2[/latex] months; b. [latex]3[/latex] years, [latex]5[/latex] months

- Sally took out a [latex]\$475,000[/latex] mortgage at [latex]2.9\%[/latex] compounded semi-annually. She repaid the mortgage with monthly payments for [latex]25[/latex] years. How much would the amortization period be shortened by if Sally made bi-weekly payments of [latex]\$1,100[/latex] instead of the monthly payments?

**Solution**[latex]2[/latex] years, [latex]4[/latex] months

- Sam received a [latex]\$280,000[/latex] mortgage at [latex]4.5\%[/latex] compounded semi-annually. She repaid the mortgage with monthly payments for [latex]20[/latex] years. How much would the amortization period be shortened by if Sam made a [latex]\$12,000[/latex] lump-sum payment at the end of year two?

**Solution**[latex]1[/latex] year, [latex]2[/latex] months

- A [latex]\$420,000[/latex] mortgage at [latex]3.47\%[/latex] compounded semi-annually is repaid with monthly payments for [latex]30[/latex] years. How much would the amortization period be shortened by if a lump-sum payment of [latex]\$20,000[/latex] is made at the end of year six?

**Solution**[latex]1[/latex] year, [latex]11[/latex] months

- Jesse received a [latex]\$570,000[/latex] mortgage at [latex]2.9\%[/latex] compounded semi-annually. He repaid the mortgage with monthly payments for [latex]25[/latex] years. How much would the amortization period by shortened by if Jesse increases his monthly payments by [latex]\$175[/latex] at the end of year five?

**Solution**[latex]1[/latex] year, [latex]7[/latex] months

- A [latex]\$190,000[/latex] mortgage at [latex]3.2\%[/latex] compounded semi-annually is repaid with monthly payments for [latex]15[/latex] years. How much would the amortization period be shortened by if the monthly payments are increased by [latex]10\%[/latex] at the end of year three?

**Solution**[latex]1[/latex] year, [latex]3[/latex] months

#### Attribution

“13.4: Special Application: Mortgages” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“13.4: Special Application: Mortgages” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.