It is often necessary to take out a mortgage when purchasing a home or property.  A mortgage is usually a long-term (roughly 25 year), general (P/Y ≠ C/Y), ordinary annuity (PMTs at END of interval).  Buyers can choose a fixed or variable rate mortgage.  A fixed rate mortgage means that the interest rate charged remains fixed for the mortgage term.  A variable rate mortgage means that the interest rate varies throughout the mortgage term. In general, the buyer (mortgage holder) is required to pay down some of the balance owing (principal) on the mortgage in addition to the interest charges each interval.

History of How Mortgages Evolved over the Years

In the 1900’s, home buyers who took out a mortgage only paid the interest owed each month.  The buyer would save up while making these interest-only payments and fully repay the mortgage when they had enough saved.^[1]

Major world events (like the first world war & great depression) changed this practice.  Large numbers of people were unable to pay their mortgages.  Because of this, buyers were then required to pay some of the balance owing (principal) each payment interval in addition to the interest owed each month.  To further protect lenders, in 1946, the “Canada Mortgage and Housing Corporation(CMHC)” was created.  The CMHC insures buyers’ mortgages if the buyer puts down less than a minimum % down payment.^[2]  This protects lenders if a buyer defaults on their mortgage (can’t pay their mortgage).

Before the 1970/80’s, buyers would be guaranteed a fixed interest rate for the entire duration of their mortgage.  This changed after the interest rate inflation in the 1970’s and 80’s (interest rates hit a record high of 21.46%).  Banks who had previously locked in interest rates with buyers were missing out on thousands of potential dollars in interest when buyer were locked in at rates as low as 6.9% and when the current interest rate was 21.46%.  After the interest rate inflation of the 1970’s and 1980’s, banks created “mortgage terms.”

Mortgage Terms

A mortgage term is the length of time your mortgage agreement and interest rate will be in effect.^[3]  Mortgage terms are, most often, 5 years in length but can vary anywhere from 6 months to 10 years in length.  If the buyer chooses a fixed-rate mortgage, they are guaranteed a fixed interest rate for the duration of the mortgage term.  After the term is ‘up’ (after the time period defined by the term has passed), the buyer negotiates a new interest rate with the lender (bank).  The buyer renews their mortgage.

Basically, a new mortgage is drawn up at the end of each term when the buyer renews their mortgage.  The buyer can pay off part (or all) of the balance owing with a lump-sum payment.  The buyer can move their mortgage to another bank. The buyer is charged interest at the new interest rate and pays this interest on the current balance (principal) owing on the mortgage.

If each term is 5 years in length, a buyer will renew their mortgage roughly five times on a 25-year mortgage.  The full length of the mortgage (ex: 25 years) is the amortization period.

Constructing Amortization Tables

Amortization is “the process of paying off debt over time with regular installments of interest and principal sufficient to repay the loan in full by its maturity date.^[4]”

We can create an amortization table (or schedule) to show the amount of principal and interest that make up each payment. The table also shows the balance owing after each payment. The table can run until the loan (or mortgage) if fully paid off or to the end of the term.

Let us now look at an example of an amortization schedule for Kerry — she just bought a car and borrowed some money from her line of credit to do so.

Example 5.10.1

Kerry purchases a new Toyota Rav4. She borrows $16,000 from her line of credit for the purchase. She is charged 3% compounded monthly on the line of credit. Kerry wants to repay her line of credit with 6 monthly payments. The first payment will be in one month. Construct an amortization schedule and use it to answer how much interest she will pay in total and what the size of her final payment will be. Assume Kerry owed nothing on her line of credit before she borrowed the $16,000.

In order to construct the amortization schedule, we need to determine the size of Kerry’s monthly payments:

Let’s round Kerry’s payment up to the next dollar^[5] which means that Kerry will pay $2,691 per month.

Because we round up Kerry’s payments, this means that she will overpay by roughly $0.95 per month. This makes her final payment slightly smaller. Let us now construct the amortization schedule to determine the size of this final payment as well as the interest paid and balance outstanding each month.

Line of Credit Amortization Table

Key Takeaways for the Above Amortization Table

• Payment Amount = PMT = $2,691 for this example
• Interest Paid (INT) = Previous Ending Balance × i
• Where i = periodic rate = ^j_m ⁄_m = ^0.03⁄₁₂ = 0.0025
• Principal Repaid (PRN) = Payment Amount − Interest Paid
• Ending Balance (BAL) = Previous Ending Balance − Principal Repaid
• Final Payment = Payment (PMT) + Final Ending Balance = $2,691+(−$5.74) = $2,685.26
• Add up all of the interest paid to calculate the total interest paid:
  Total Interest = $40 + $33.37 + $26.73 + $20.07 + $13.39 + $6.70 = $140.26
• You could also construct this table in Excel (click here to download the Excel file).

Conclusion: Kerry will pay $140.26 in interest total and make a final payment of $2,685.26 to pay off her line of credit.

Amortization Using AMRT in the BAII Plus

It can take a long time to construct an amortization table, especially for long-term loans. You can instead construct the table in Excel or use the BAII Plus’ AMRT (amortization) menu. In order to access the AMRT menu in your BAII Plus, you need to hit 2ND PMT after you have already entered all values in the TVM keys and have rounded up your payment (PMT) and re-entered it into your calculator as a negative value. These steps are written out below:

Steps to Using the AMRT Menu in Your BAII Plus

1. Compute the missing value in the TVM keys (ex: PMT)
2. Input the rounded up payment value. Then make it negative and re-enter it using: + | − PMT
3. Hit 2ND PMT to enter the AMRT menu
4. Enter in a value for P1 and hit ENTER   ↓  
5. Enter in a value for P2 and hit ENTER   ↓  
6. Scroll through the AMRT menu using   ↑      ↓  

In the above steps, we did not explain what P1 and P2 mean. Let’s now make sense of these values as well as the rest of the values given by the AMRT menu.

Understanding the Values in the AMRT Menu:

• P1: Starting payment in period in question
• P2: Ending payment in period in question
• BAL: Outstanding Balance at end of period in question
• PRN: Principal Repaid during period in question
• INT: Interest Paid during period in question

We have not yet clearly defined what the “period in question” means. This is because this period can vary. For monthly payments, the period in question is the starting and ending month numbers. Some examples are given below for P1 and P2 values.

Examples of “Periods in Question” for Monthly Payments

1. The first year of a mortgage ⇒ P₁=1, P₂=12 (there are 12 months in the first year)
2. The first month of a mortgage ⇒ P₁=1, P₂=1 (period is just month 1)
3. The third year of a mortgage ⇒ P₁=25, P₂=36 (months 25 to 36 are in the third year)
4. The third month of a mortgage ⇒ P₁=3, P₂=3 (just period 3)
5. The first three years of a mortgage ⇒ P₁=1, P₂=36 (payments 1 to 3×12=36)

Example 5.10.2

Maksim purchases an apartment in New Westminster. He pays $600,000 less a 20% down payment. He takes out a 25-year mortgage with a 5-year term. He is charged 2.95%, compounded semi-annually. He makes monthly payments with the first payment in one month. How much interest will he pay during the first 5-year term? Assume that his payments are rounded up to the next dollar.

Step 0: Determine the amount Maksim borrows (PV):

[latex]\begin{align*} \textrm{Amount Borrowed} &= \textrm{Price} - \textrm{Down Payment} \\ &= \$600,000 - \$600,000 \times 0.20 \\ &= \$600,000\times (1- 0.20) \\ &= \$480,000 \end{align*}[/latex]

Step 1: Determine the size of Maksim’s monthly mortgage payments:

Step 2: Round up the payment and re-enter as a negative value: 2260 + | − PMT

Step 3: Access the AMRT menu: 2ND PMT

Step 4: Input P₁: 1 ENTER   ↓  

Step 5: Input P₂: 60 ENTER   ↓  

Step 6: Scroll down to INT:   ↓     ↓     ↓  

Conclusion: Maksim will pay $65,436.89 in interest during the first 5 years of his mortgage.

Wondering why we used P₁=1 and P₂=60? There are 60 months in the first 5 years, starting with month one and ending with month 60. Also see the table in the section below below.

P1 & P2 Values for First 5 years of Monthly Payments

We want to calculate the interest for the first 5 years of the mortgage (the first term). The first month in this time-period is month 1 (the start of the mortgage). The last month will be month 60 (=5×12). See the table below for the month numbers for the first 5 years.

Example 5.10.3 — How much of the first mortgage payment will be interest?

Step 0—2: Make sure the values from Example 2 entered into the TVM keys (N, I/Y, PV, PMT, FV) in your BAII Plus.

Step 3: Access the AMRT menu: 2ND PMT

Step 4: Input P₁ (the first payment ‘starts’ in month 1): 1 ENTER   ↓  

Step 5: Input P₂ (the first payment ‘ends’ in month 1): 1 ENTER   ↓  

Step 6: Scroll down to INT:   ↓     ↓     ↓  

Conclusion: Maksim will pay $1,172.82 in interest during the first month of his mortgage.

Example 5.10.4— How much will the balance owing be reduced by with the first payment?

In this case, we want to calculate the principal repaid (this is the amount we reduce the balance owing by with each payment that we make). Make sure all values from Example 2b are still in your calculator (Steps 0 to 5). Just scroll back up  ↑  to PRN.

Conclusion: Maksim will reduce his balance owing by $1,087.19 with his first mortgage payment.

Example 5.10.5— How much interest does Maksim repay in the first year?

There are 12 months of payments in the first year, starting at payment 1:

P₁ = 1
P₂ = 12
INT = $13,896.99

Conclusion: Maksim will pay $13,896.99 in interest in the first year.

Example 5.10.6 — How much principal does Maksim repay in the fifth year?

The fifth year starts at month 49 and ends at month 60:

P₁ = 49
P₂ = 60
PRN = $14,866.29

Conclusion: Maksim will reduce his balance owing by $14,866.29 in the fifth year.

Example 5.10.7 — How much Maksim still owe at the end of five years?

If we want the balance owing, it only matters what is entered into P2. This is because BAL (balance) is the amount owing at the end of payment P2. For this reason, we can just leave the values from Example 2e in the calculator and scroll to BAL:

P₁ = 49
P₂ = 60
BAL = $409,836.89

Conclusion: Maksim will owe $409,836.89 at the end of five years.

Renewing Mortgages

At the end of a mortgage term, the mortgage holder renews their mortgage (or refinances the mortgage if they borrow more money).

When the mortgage holder renews their mortgage, their terms and interest rate will most likely be changed. Use this new rate and use the number of years remaining to determine the size of the new mortgage payments.

Example 5.10.8

Let us assume Maksim has made 5 years of mortgage payments and it is now time for Maksim’s to renew his mortgage.  Maksim renews his mortgage for another 5 years at 3.5% compounded semi-annually.  What is the size of Maksim’s new mortgage payments? Round up to the next dollar.

Conclusion: Maksim will pay $2,372 per month (round the payments up).

Calculating the Final Payment

The final payment using the BAII Plus AMRT method can be calculated the same way as when we using the amortization table (see Example 1). The only differences are that we enter the final payment number into P2 and scroll down to BAL to determine the final balance owing. We then use the same formula as before:

[latex]\textrm{Final Payment} = \textrm{Regular Payment Size (PMT)} + \textrm{Final Ending Balance}[/latex]

Note: the ending balance will be negative. That means that when we add that negative number to the regular payment size, the payment size drops in value.

Example 5.10.9

Let us continue on with Example 5.10.9. Let us assume Maksim continues to pay 3.5% compounded semi-annually for the entire 20 years remaining in his mortgage. What will be the size of Maksim’s final payment if this were true?

Step 1: Make sure the values from Example 3 entered into the TVM keys.

Step 2: Round up the payment and re-enter as a negative value: 2372 + | − PMT

Step 3: Access the AMRT menu: 2ND PMT

Step 4: Input P₁ (input any value up to 240): 1 ENTER   ↓  

Step 5: Input P₂ (final payment is the 240^th payment): 240 ENTER   ↓  

Step 6: Scroll down to BAL:   ↓  

Step 7: Calculate the final payment size using BAL=−147.20:

[latex]\textrm{Final Payment} = \$2372 + (-$147.20) = $2,224.80[/latex]

Conclusion: Maksim will make a final payment of $2,224.80 at the end of 20 years.

Calculating the Interest Charged Using the Formula

There are two ways to calculate the total interest charged on a mortgage. We can calculate the difference between the money out and money in or we can use the AMRT menu and read off the INT values. Let us first step through taking the difference between money out and in calculation:

[latex]\textrm{Interest Charged} = \textrm{Money Out} - \textrm{Money In} = \textrm{\$OUT} – \textrm{\$IN}[/latex]

To determine $ OUT, where be sure to add up ALL payments and be careful of the final payment — it is often smaller than the rest:

[latex]\textrm{\$ OUT} = \textrm{Sum of All Mortgage Payments}[/latex]

To determine $ IN, total all money borrowed. If the mortgage holder borrows more money at some point during the mortgage (if they refinance), then also include that amount in the $ IN calculation:

[latex]\textrm{Money In (\$ IN)} = \textrm{Total Amount Borrowed}[/latex]

Let us now determine how much interest Maksim will be charged on this mortgage.

Example 5.10.10

Let us continue on with this example. Let us assume Maksim continues to pay 3.5% compounded semi-annually for the entire 20 years remaining in his mortgage. If this is true, how much interest does Maksim pay in total on his mortgage?

Let us first calculate the money out ($ OUT):

[latex]\begin{align*} \textrm{Money Out (\$ OUT)} &= \textrm{Sum of All Mortgage Payments}\\ &= \$2,260 \times 60 + \$2372 \times 239 + \$2,224.80 \\ &= \$135,600+\$566,908+\$2,224.80=\$704,732.80 \end{align*}[/latex]

Next, let’s determine the money in ($ IN):

[latex]\begin{align*} \textrm{Money In (\$ IN)} &= \textrm{Total Amount Borrowed}\\ &= \$480,000 \end{align*}[/latex]

Taking the difference between the money out and in gives:

[latex]\textrm{Interest Charged} = \$704,732.80 – \$480,000 = \$224,732.80[/latex]

Conclusion: Maksim will be charged $224,732.80 in interest over the 25 years.

Calculating the Interest Charged Using the AMRT Menu

Let us now step through how to use the AMRT menu results to calculate the total interest charged on a mortgage.

For each term in the mortgage (or for each period where the interest rate is fixed), calculate the total interest paid during that term by doing the following:

P₁ = 1
P₂ = Final Payment # in term
INT = Total interest paid during that term

Redo this calculation for each term in the mortgage and add up all of these values to determine the total interest paid over the entire mortgage. Let us revisit Maksim’s mortgage a final time in this section to understand how to perform this calculation.

Example 5.10.11 — Calculate the total interest charged using the AMRT menu

We have already calculated the total interest charged for the first term in Maksim’s mortgage. We found that Maksim paid $65,436.89  in interest during the first 5 years of his mortgage.

To determine the amount of interest Maksim will pay in the remaining 20 years^[7], let us look back at Example 4 and do the following:

1. Make sure the values from Example 4 are still in the TVM keys (they should be)
2. Access the AMRT menu: 2ND PMT
3. Make sure P₁=1 (it should still be 1)
4. Make sure P₂=240 (it should still be 240)
5. Scroll down to INT:   ↓     ↓     ↓    = $159,295.91

Use this interest amount as well as the $65,436.89 to determine the total interest charged:

Total Interest Charged = $65,436.89 + $159,295.91 = $224,732.80

Conclusion: Maksim will be charged $224,732.80 in interest over the 25 years (the same amount as calculated in Example 5).

Your Own Notes

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