Introduction To Interest Rates And Mortgage Mathematical Calculations

INTRODUCTION TO
Interest Rates and Mortgages
Chapter 14

Interest Rates & Mortgages
➢ Interest Rate Conversions
➢ Constant Payment Mortgages
➢ Outstanding Balances
➢ Accelerated Bi-Weekly Payments
➢ Principal & Interest Splits

Constant Payment Mortgages
Constant payment mortgage calculations mainly are used in finding out, the size of
payments on a particular loan, the balance owing on an existing loan, or the amount
of a loan that a payment will support.
There are generally 4 financial components in constant payment mortgage loans:
1. Loan Amount
2. Amortization Period
3. Nominal Rate of Interest
4. Payment

CONSTANT PAYMENT MORTGAGES - Components
1. Loan Amount: The “Face Value” of the loan. The amount the borrower agrees to
repay. Otherwise known as the “Present Value” pertaining to the payment agreed
upon
2. Amortization Period: The “length” of the loan. What the payments are determined
upon
3. Nominal Rate of Interest: The interest rate the loan is written at - nominal interest
rate must match the frequency of the payments
4. Payment: The amount of principal and interest paid over the life of the loan or
“Amortization Period” that will eventually pay the loan off in full

Interest Rate Conversions
Does the compounding frequency given in the question match the payment frequency?
If NO, an Interest Rate Conversion is needed
This is common because of the Interest Act
- The Interest Act requires lenders to disclose the interest rates they charge as
a J1, or J2 which is a”yearly” or “semi annually” compounding
However - most people make payments in different ways. Most commonly
“monthly” especially when pertaining to mortgage payments

Rate Conversions - The Formula
1st - Input the amount of interest
2nd - Press “shift” NOM
3rd - Input the number of compounding periods
4th - Press “shift” P/YR
5th - Press “shift” EFF
6th - Input the new number of compounding periods
(must match the frequency they are paying, monthly, quarterly, etc.)
7th - Press "shift” P/YR
8th - Press “shift” NOM

Rate Conversions - The Formula - Example
Convert J2 = 4.5% to J12
1) 4.5 “shift” NOM% = 4.5
2) 2 “shift” P/YR = 2
3) “shift” EFF% = 4.550625
4) 12 “shift” P/YR = 12
5) “shift” NOM% = 4.458383
J12 =
4.458383

Constant Payment Mortgages
- Formulas and Examples
Different examples to go through
➢ Finding the “Payment”
➢ Finding the “Present Value”
➢ Finding the “Interest Rate”
➢ Finding the “Amortization Period”

1. Finding the “Payment”
Ex. Calculate the monthly payment required for the following mortgage:
Principal of $40,000; 14% Interest per annum, compounded semi-
annually;
amortization period of 20 years
Breakdown:
Loan (PV) = $40,000,
Interest J2 = 14%
Monthly Payments
Amortization 20 Yrs
Loan will full be paid off after 20
years

1 . Finding the “Payment”
Principal of $40,000; 14% Interest per annum, compounded semi-
annually;
Breakdown:
Loan (PV) = $40,000,
Interest J2 = 14%
Monthly Payments
Amortization 20 Yrs
years
Q. Does the payment frequency match the interest rate compounding frequency?
- If not we need to do a rate conversion

1. Finding the “Payment”
Principal of $40,000; 14% Interest per annum, compounded semi-annually;
Breakdown:
Loan (PV) = $40,000, Interest J2
= 14%
Monthly Payments
Amortization 20 Yrs
Loan will full be paid off after 20 years
Step 1. Rate Conversion: J2 to a J12
14 * NOM
2 * P/YR
* EFF%
12 *P/YR
*NOM
Step 2. Complete the Payment Calculation
40 000 PV
20 x 12 = 240 N
0
FV

2. Finding the “Present Value”
EX. An investor wants to decide whether to buy a mortgage that calls for monthly
payments of $390 for 20 years. If the investor can earn j2 = 8%, at what price should
the mortgage be purchased?
Breakdown:
Interest J2 = 8%
Monthly Payments $390
Amortization 20 yrs
years

EX. An investor wants to decide whether to buy a mortgage that calls for monthly
payments of $390 for 20 years. If the investor can earn j2 = 8%, at what price should
the mortgage be purchased?
Breakdown:
Interest J2 = 8%
Monthly Payments $390
Amortization 20 yrs
years

EX. An investor wants to decide whether to buy a mortgage that calls for monthly payments of
$390 for 20 years. If the investor can earn j2 = 8%, at what price should the mortgage be
purchased?
Breakdown: Interest J2 = 8% Monthly
Payments $390
Amortization 20 yrs
years
8 * NOM
2 * P/YR
* EFF%
12 *P/YR
*NOM
Step 2. Complete the PV (Present Value) Calculation
20 x 12 = 240 N
0
FV
390 +/- PMT
47,081.1122383 PV

3. Finding the “Interest Rate”
Ex. A private investor expects to receive $281.72 per month for a
period of 17 years as a result of a mortgage loan she has just advanced.
Calculate the investor’s expected yield (expressed as a nominal rate with
semi-annual compounding) on her investment if the loan was for $23,250.
Breakdown:
Loan (PV) = $23,250
Monthly Payments $281.72
Length of loan 17 Yrs
Loan will fully be paid after 17 years
Express yield with semi-annual
compounding

Ex. A private investor expects to receive $281.72 per month for a
period of 17 years as a result of a mortgage loan she has just advanced.
Calculate the investor’s expected yield (expressed as a nominal rate with
semi-annual compounding) on her investment if the loan was for $23,250.
Breakdown:
Loan (PV) = $23,250
Monthly Payments $281.72
Length of loan 17 Yrs
Loan will fully be paid after 17 years
Express yield with semi-annual
compounding

Ex. A private investor expects to receive $281.72 per month for a period of 17 years as
a result of a mortgage loan she has just advanced. Calculate the investor’s expected yield
(expressed as a nominal rate with semi-annual compounding) on her investment if the loan
was for $23,250.
Breakdown: Loan (PV) = $23,250 Monthly Payments
$281.72
Length of loan 17 Yrs Loan will fully
be paid off after 17 years
Express yield with semi-annual compounding
Step 1. Find the Nominal Interest rate
23 250 PV
0
FV
281.72 PMT
17 x 12 = 204 N
12
Step 2. Rate Conversion J12 to J2
12.898410 *NOM (already entered)
12 *
P/YR (already entered)
* EFF%
2 *P/YR
*NOM

4. Finding the “Amortization Period”
Ex. A Mortgage in the amount of $150,000 requires the buyer to make
payments of $1,250 per month for as long as necessary to fully amortize
the loan at 8% per annum, compounded semi-annually. How many full
payments of $1,250 will be required?
Breakdown:
Loan (PV) = $150,000
Monthly Payments $1,250
J2 = 8% Loan will fully
be paid off at the end

Ex. A Mortgage in the amount of $150,000 requires the buyer to make
payments of $1,250 per month for as long as necessary to fully amortize
the loan at 8% per annum, compounded semi-annually. How many full
payments of $1,250 will be required?
Breakdown:
Loan (PV) = $150,000
Monthly Payments $1,250
J2 = 8% Loan will fully
be paid off at the end

Ex. A Mortgage in the amount of $150,000 requires the buyer to make payments of
$1,250 per month for as long as necessary to fully amortize the loan at 8% per
annum, compounded semi-annually. How many full payments of $1,250 will be
required?
Breakdown: Loan (PV) = $150,000 Monthly
Payments $1,250
J2 = 8% Loan will fully be
paid off at the end
8 * NOM
2 * P/YR
* EFF%
12 *P/YR
*NOM
7.869836 = J (Monthly
Step 2. Complete the “N” (Amortization Period) Calculation
150 000 PV
0 FV
-1250 PMT
236.566775 N
236 Full Payments will be Required

Outstanding Balances & Introducing Terms
While mortgage payments are calculated using the amortization period, the actual
length of the mortgage contract may be different than the amortization period
The Length of a mortgage contract is referred to as a “Term”. The life of the
mortgage loan is split up into “Terms” because of the sometimes long
Amortization Periods that come with Mortgage Loans.
Due to these Terms, it is important to be able to calculate an Outstanding
Balance, as even though the Term is over, it does not mean that the loan is paid
in full - or in other words “Fully Amortized”

Outstanding Balances & Introducing Terms
The terminology we use is as follows;
Fully Amortized Loans: When the mortgage term and amortization period are
the same length of time, the mortgage is referred to as being Fully Amortized
Partially Amortized Loans: When the mortgage term is shorter than the
amortization period, the mortgage is referred to as being Partially Amortized

Calculating Outstanding Balances
When you see a Term in a question that is shorter than the amortization period,
we know we have to calculate the outstanding balance.
To Calculate the outstanding balance we first have to find the payments based
on the fully amortized loan and then re-entering the rounded payment back into
the calculator.
If the question tells you to round to a specific number, you must do that and re-
enter the payment. If no instruction to round is given, round to the nearest 2
decimal points, and re-enter. DON'T FORGET TO ENTER IN AS A NEGATIVE

Calculating Outstanding Balances - Example
Ex. A borrower is arranging a mortgage with Nicety Finance Company. The loan
amount is $175,000, the interest rate is 4.5% per annum, compounded semi-
annually, the amortization period is 20 years, and the contractual term is 2 years. If
payments are made monthly and rounded up to the next higher $10, calculate the
outstanding balance at the end of the loan term.
Breakdown:
Loan (PV) = $175,000,
Interest J2 = 4.5%
Monthly Payments (Nxt high$10)
Amortization 20 Yrs

Ex. A borrower is arranging a mortgage with Nicety Finance Company. The loan
amount is $175,000, the interest rate is 4.5% per annum, compounded semi-
annually, the amortization period is 20 years, and the contractual term is 2 years. If
payments are made monthly and rounded up to the next higher $10, calculate the
outstanding balance at the end of the loan term.
Breakdown:
Loan (PV) = $175,000,
Interest J2 = 4.5%
Amortization 20 Yrs
Term 2 Yrs

Breakdown: Loan (PV) = $175,000, Interest J2
= 4.5%
Amortization 20 Yrs
Term 2 Yrs
4.5 * NOM
2 * P/YR
* EFF%
12 *P/YR
4.458383 *NOM
Step 2. Finish payment calculation
175 000 PV
20 x 12 N
0 FV
Step 3. Re-Enter Payment & Calculate Outstanding Balance
based on length of Term
1,110 +/- PMT
2 x 12= 24 Input shift AMORT
= = =
-163,479.729771 FV
Step 4. Round answer to the nearest 2 decimal places

Accelerated Bi-Weekly Payments
There are differences between payment options for mortgages
Constant Monthly Payments: Equal Payments that are paid once a month
Bi-Weekly Payments: Constant Payments that are paid every two weeks
Accelerated Bi-Weekly Payments: Constant Payments that are equal to half of the
regular monthly payment and are paid every two weeks
Accelerating payments is a very effective way to pay off a mortgage loan faster and to
reduce amount if interest paid. It also works with many peoples pay structure of
receiving income every 2 weeks

Accelerated Bi-Weekly Payments
Accelerated Bi-Weekly Payments: Constant Payments that are equal to half of the
regular monthly payment and are paid every two weeks
To calculate the Accelerated Bi-Weekly payment, you must first find the Constant
Monthly Payment based on the Fully Amortized Loan and divide it in half.

Accelerated Bi-Weekly Payments - Example
Ex. A mortgage loan has a face value of $350,000, an interest rate of j2 = 5.5%,
an amortization period of 20 years, a term of 3 years, and an option to make
accelerated biweekly payments. What is the amount of the accelerated biweekly
payment rounded up to the next highest dollar?
Breakdown:
Loan (PV) = $350,000,
Interest J2 = 5.5%
Biweekly Payments
Amortization 20 Yrs
Term 3 Yrs

Ex. A mortgage loan has a face value of $350,000, an interest rate of j2 = 5.5%,
an amortization period of 20 years, a term of 3 years, and an option to make
accelerated biweekly payments. What is the amount of the accelerated biweekly
payment rounded up to the next highest dollar?
Breakdown:
Loan (PV) = $350,000,
Interest J2 = 5.5%
Biweekly Payments
Amortization 20 Yrs
Term 3 Yrs

Ex. A mortgage loan has a face value of $350,000, an interest rate of j2 = 5.5%, an
amortization period of 20 years, a term of 3 years, and an option to make accelerated biweekly
payments. What is the amount of the accelerated biweekly payment rounded up to the next
highest dollar? Breakdown:
Loan (PV) = $350,000,
Interest J2 = 5.5%
Biweekly Payments (Nxt high$1)
Amortization 20 Yrs
Term 3 Yrs
NOTE: To calculate the Accelerated Bi-Weekly payment, you must first find the
Constant Monthly Payment based on the Fully Amortized Loan and divide it in half.
5.5 * NOM
2 * P/YR
* EFF%
12 *P/YR
*NOM
Step 2. Complete the Payment Calculation
350,000 PV
20 x 12 = 240 N
0
FV
-2,395.36951 PMT
2,395.36951 ÷ 2 =

Principal and Interest Splits
It is sometimes necessary to calculate either the Interest or the Principal of a Constant
Payment Mortgage Loan at any given time during the life of the loan.
It is good information to know if thinking of re-signing the loan, looking to see just
where you are in regards of repayment, or because interest on payments can
sometimes be deducted as an expense for income tax purposes
- To find the Principal and Interest at any given time in the Mortgage Loan life, we
first have to find the Payments based on the fully amortized loan and then re-
entering the rounded payment back into the calculator.
- At that time we can then use our financial calculator to find the Principal/Interest

Principal and Interest Splits - Calculator Keys
SHIFT Key : To access the Orange keys needed
in our Financial Calculator
INPUT Key : To Input the value of time into our
AMORT key
AMORT Key : The key that calculates the
Amortization of each equation. - Calculates
Outstanding Balance, Interest and Principal
separately for our knowledge and use

Principal and Interest - Example
Ex. A borrower has arranged a $159,900 mortgage at j12 = 12% with a 25-year
amortization, 5-year term, and monthly payments. If all payments are paid when due,
how much principal was paid off during the 5-year term?
Breakdown:
Loan (PV) = $159,900,
Interest J12 = 12%
Monthly Pmys
Amortization 25 Yrs
Term 5 Yrs

amortization, 5-year term, and monthly payments. If all payments are paid when due,
how much principal was paid off during the 5-year term?
Breakdown:
Loan (PV) = $159,900,
Interest J12 = 12%
Monthly Pmts
Amortization 25 Yrs
Term 5 Yrs

amortization, 5-year term, and monthly payments. If all payments are paid when due, how much
principal was paid off during the 5-year term?
Breakdown: Loan (PV) = $159,900, Interest
J12 = 12%
Monthly Pmts
Amortization 25 Yrs
Term 5 Yrs
Step 1. Find the Payment based on the Full Loan, round
and re-enter payment into PMT key
159,900 PV
25 x 12 = 300 N
0
FV
12 *
Step 2. Find the amount of principal that has been paid
off during the 5-year term
1
INPUT
60 *
AMORT

Practice the 1000 questions
Chapter 14
Thank you!

Introduction To Interest Rates And Mortgage Mathematical Calculations

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