UNIT 2: BORROWING MONEY
Unit 2 – Lesson 1 – Analyzing Loans
Date:
Recall investing money:
- Investing means to lend your money to a borrower (a government, bank, organization or person)
- The borrower gives you ‘interest’
- You deposit a lump sum of money at the start of the term, then redeem that original lump sum, plus interest, at
maturity (simple interest or compound interest) OR you deposit regular payments during the duration of the
investment, then redeem all your payments plus the interest earned on each payment, at maturity (annuity)
Loans:
- You are the borrower; you borrow a sum of money (called a loan) from a lender (a government, bank,
organization or person)
- You pay the lender ‘interest’
- You receive a lump sum of money at the start of the loan’s term, and
o you repay the entire original amount, plus interest, at the end of the loan’s term (simple interest)
o OR you repay the original amount, plus interest, by making regular payments. (annuity)
- Banks will generally ask for an asset to hold as security against the repayment of a loan - called collateral. If you
do not pay back your loan under the loan’s conditions, the bank will have the right to take the collateral asset.
Types of loans:
SIMPLE INTEREST
LOAN
Clue words in the
problem:
- you will see the
words ‘simple interest’
somewhere in the
description of the
loan.
How to solve:
use the formula
= + A: value at maturity;
the amount you pay at
the end of the term
(the principal plus
interest)
P: the amount you
borrow at the
beginning of the term
r: yearly interest rate
t: term of the loan, in
years
COMPOUND INTEREST LOAN
Clue words in the problem:
- you will see words such as ‘compound interest’ or
‘interest is compounded monthly’
- compounded yearly or annually: 1 time per year
- compounded semiannually: 2 times per year
- compounded quarterly: 4 times per year
- compounded monthly: 12 times per year
- compounded weekly: 52 times per year
- compounded daily: 365 times per year
How to solve:
1) use the formula = (1 + )
A: the amount you pay back at the end of the term
(the principal plus interest)
P: the amount you borrow at the beginning of the
term
n: total number of compounding periods
i: interest per compounding period
Limitations of the formula: cannot solve for n
2) use the graphing calculator
N= total number of compounding periods
I%= yearly interest rate, in percent form
PV= principal amount (P); amount borrowed
PMT=0
FV= future value (A); amount you pay back at
maturity
P/Y= make it the same number as C/Y
C/Y= # of compounds per year
PMT: END
ANNUITY LOAN
Clue words in the problem:
- you will see words like ‘regular payments’ or
‘monthly payments’ or ‘monthly installments’
- you will also see ‘compounded yearly’ or
‘compounded monthly’ etc
How to solve:
1) use the formula = ×
()
PV: The lump sum you borrow at the start of the
loan
R: amount of each regular payment; each
payment is composed of ‘principal’ and ‘interest’
n: total number of compounding periods
i: interest per compounding period
Limitations of the formula:
- cannot solve for i or n; can only use it to solve for
PV or R.
- compounding frequency and payment frequency
must be the same
2) use the graphing calculator
N= total number of payments
I%= yearly interest rate, in percent form
PV= principal amount; amount borrowed
PMT= regular payment (R)
FV= 0
P/Y= # of payments per year
C/Y= # of compounds per year
PMT: END
Example 1: Alex needs $4,000 right now to pay for school tuition. Analyze his loan options. Calculate the amount of
interest he will pay in each scenario to recommend the best loan option.
a) A loan at 5% simple interest, to be paid back with one single payment in 4 years.
b) A loan at 5% compounded daily, to be paid back in full in 4 years.
c) A loan at 5% compounded monthly, to be paid back in monthly payments over a 4 year term.
Class work/HW: p. 92 #1, 2, 3, 4a, 5, 6, 9
Unit 2 – Lesson 2 – Amortization tables
Amortization table:
- In an annuity loan, each regular payment is made up of some interest and some principal
- The make-up of each payment is different, even though all payments are equal. The reason for this is, as the
borrower makes payments, the money owing decreases, thus the amount of interest for each payment period
decreases with time.
- An amortization table lists regular payments of a loan and shows how much of each payment goes toward the
interest charged and the principal borrowed, as the balance of the loan is reduced to zero.
Example 1: Tina wants to borrow $1,000 to purchase a bike. She can afford to pay back $300 per month. The bank offers
her a line of credit at 12% compounded monthly.
a) How long will Tina take to pay back the loan if she makes regular monthly payments of $300?
N
I (%)
PV
PMT
FV
P/Y
C/Y
END/BEGIN
b) Set up an amortization table showing the interest-principal composition of each payment.
Payment Payment Interest Paid ($)
Period
($)
(month)
0
1
300
2
300
3
300
Principal Paid ($)
Balance Owing ($)
1,000
4
Totals
The graphing calculator does not have an amortization table, though there are websites where you can find amortization
tables. They work similarly to the TVM solver – you enter all the required parameters (amount of loan, interest rate etc),
and the application creates a similar table to the one above.
The graphing calculator does however have a few functions that give you specific values in the amortization table.
To access the ‘extra’ Finance functions:
• First, enter all the known data in the TVM solver, and solve for the unknown value (the usual routine)
• Second, press APPS – Finance, then scroll down through the list to find the function you want.
9: bal (
This function returns the ‘balance owing’ amount for a specified payment number. You must enter
the payment number in brackets.
bal(period number)
Ex: bal (3) will return the balance owing after 3 payments.
This function does not work with partial payments!
: ∑ (
This function adds up the principal portions of the specified payments. You must enter the start
and end periods that you want to add up, with a comma in between. ΣPrn(Begin Period, End Period)
Ex: ∑ (1,3) will return the sum of the principal portion of the first three payments. (payment 1
through payment 3)
Ex2: ∑ (3,3) will return the principal portion of just the third payment.
This function does not work with partial payments!
A: Σ Int (
This function adds up the interest portions of the specified payments. You must enter the start
and end periods that you want to add up, with a comma in between. ΣInt(Begin Period, End Period)
Ex: ∑ (1,3) will return the sum of the interest portions of the first three payments. (payment 1
through payment 3)
Ex2: ∑ (4,4) will return the interest portion of just the fourth payment.
This function does work with partial payments!
Example 2: Calculate the total amount of interest paid on Tina’s loan, using the Σ Int function.
On your own: Calculate the number of payments required, then create an amortization table for the following loan: A
loan of $2,100 at 3% compounded monthly, to be paid back with monthly payments of $400.
a) Number of payments:
N
I (%)
PV
PMT
FV
P/Y
C/Y
END/BEGIN
b) Amortization table.
Payment Payment Interest Paid ($)
number ($)
0
1
$400
Totals
Class work/HW: p.93 #7, 11
Principal Paid ($)
Balance Owing ($)
1,000
Unit 2 – Lesson 3 – Mortgages
Mortgage:
-
-
A mortgage is a loan for the purchase of real estate with the real estate purchased used as collateral.
For the purchase of real estate you are required (by law) to pay a minimum ‘down payment’. The down payment is
calculated as a percentage of the real estate value. Currently, the minimum down payment in Canada is 5% of the
real estate value. This means home buyers must pay at least 5% of the real estate value up front, and borrow the
remaining money (the mortgage) needed to pay for the real estate.
The mortgage is generally paid with monthly payments, over a many, many years. The length of your mortgage loan
is called the ‘amortization period’. Currently, Canada’s maximum amortization period is 25 years.
A larger down payment reduces the size of your mortgage, and, therefore reduces the monthly payment and
interest you will pay over the life of the mortgage
Example 1: Jose is negotiating with his bank for a mortgage on a house. He has decided that he will make a 10% down
payment on the purchase price of $225,000. The bank is offering Jose a mortgage loan for the balance at 3.75%,
compounded semi-annually, with a term of 25 years and with monthly mortgage payments.
a)
b)
c)
d)
e)
How much is the down payment? How much will Jose have to borrow?
How much will each payment be?
How much interest will Jose end up paying by the time he has paid off the loan, in 25 years?
What is the interest – principal composition of the first mortgage payment?
What is the balance owing after 12.5 years?
On your own: Samantha is purchasing a new home, with a purchase price of $400,000. She will make a down payment
of 15%. She has signed a mortgage for the rest of the cost, at 3% compounded semi-annually, with monthly payments
over the next 20 years.
a)
b)
c)
d)
e)
f)
How much is the down payment?
How much will Samantha have to borrow?
How much will each payment be?
How much interest will she end up paying in total over the life of the mortgage?
What is the interest – principal composition of the first mortgage payment?
What is the balance owing after 10 years?
Class work/HW: Homework worksheet
MA12 F - UNIT 2 – LESSON 4 – SOLVING PROBLEMS INVOLVING CREDIT
Intro:
- Some common forms of credit that can be used to make purchases or acquire cash include: bank loans, lines of
credit, credit cards, payday loans, and dealership or in-store financing.
- Some loans have ‘structured’ payments. For example, a one-time lump sum paid at the end of the term, or equal
regular payments made at equal intervals. The lenders prefer these types of payments because it makes it easy
for them to predict future cash flows.
- Some loans have ‘flexible’ payments. For example, credit cards and lines of credit allow the borrower to make
payments of any sum, at any time. This type of payment arrangement is beneficial for the borrower. However,
the disadvantage is that the interest rates are usually much higher than loans with structured payments.
Credit cards
- The borrower borrows as much as they want, whenever they want (up to a maximum ‘limit’)
- The borrower pays back as much as they want, whenever they want (but they must pay a minimum monthly
payment)
- Interest charged is very high (about 19%)
- The interest is usually compounded daily
- If the borrower pays off the entire debt before the end of the month, they don’t get charged any interest. (credit
cards can serve as free short term loans!)
- The full cost of borrowing should be considered before making a decision about using a credit card. This includes
the total interest charged, as well as the total payments and the time it will take to pay off the balance.
Credit card incentives
- Credit card companies compete for your business by offering incentives:
o A cash ‘rebate’ when you first sign up with the credit card company
o Reward ‘points’ that you accumulate over time, then can use to purchase from a list of items
o Yearly percentage cash rebates (usually 1% or 2% of all purchases made with the credit card)
o Lower interest for a yearly fee
o More reward points for a yearly fee
- When deciding to use a credit card to borrow money, these incentives should also be considered in the cost of
borrowing. For example, a lower interest can reduce the interest amount, but a yearly fee will increase the
borrowing cost.
More details about credit cards:
- Credit cards have a limit, which is the maximum amount you can borrow. The credit limit varies from person to
person, based on their credit history and credit rating.
- You can use a credit card to get a ‘cash advance’. Cash advances are treated differently than a regular credit
card purchase transaction: they usually have a greater interest rate, and they don’t have a period in which no
interest is charged.
Consolidating debt:
- Consolidating debt means to take out one loan (at a lower interest rate) to pay off many other loans that have
higher interest.
- This is done to reduce the interest payments, as well as for the convenience of paying one payment only instead
of many different loan payments.
Bank of Canada prime rate:
- The prime rate is an interest rate value set by Canada’s central bank, which other financial institutions use to set
their interest rates. This interest rate is also used for transactions among banks.
- The Bank of Canada prime rate is adjusted every quarter, if needed, based on the health of the economy.
- Currently, the Bank of Canada prime rate is 1%.
Prime rate:
- The term ‘prime rate’ could also refer to a reference rate used by banks, for transactions with customers.
- Currently, the customer prime rate used by most banks in Canada is 3%.
- Loan interest rates will often be stated as ‘prime + a fixed value’. For example, ‘prime plus 5’ means that your
loan will have an interest rate of prime rate (3%) plus 5% - so your interest rate would be 8%. If the prime rate
changes, then so does your loan interest rate.
- Many mortgages, lines of credit and student loans use this type of interest calculation.
Lines of credit:
- A line of credit has a lower interest rate than most credit cards. Because of this, a line of credit can be useful for
consolidating debt.
- As with a credit card, a line of credit allows for flexibility in how the loan is paid back, as long as the minimum
payment is made.
Payday loans:
- Short term loans for borrowers who need money quickly (thus do not have time to apply for a different type of
loan), and who are able to pay the loan back with their next paycheque.
- Payday loans have very high interest rates.
Example 1: Jay wants to purchase a new sound system, which costs $2,623.95, including taxes. He does not have any
money saved up. Jay has to buy it on credit and has two options:
-
Use his new bank credit card, which has an interest rate of 14.5%, compounded daily. Because this credit card is
new, he has no outstanding balance from the previous month.
Apply for the store credit card, which offers an immediate rebate of $100 on the price, but has an interest rate
of 19.3%, compounded daily.
Jay is confident that he can make regular monthly payments of $110. Which credit card is the better option?
On your own:
Nicki wants to be debt-free in 5 years. She has two credit cards on which she makes monthly payments:
-
Card A has a balance of $2,436.98 and an interest rate of 18.5%, compounded daily.
Card B has a balance of $3,043.26 and an interest rate of 19%, compounded daily.
Nicki has qualified for a line of credit at her bank with an interest rate of prime plus 6.6%, compounded monthly, and a
credit limit of $6000. She plans to pay off both credit card balances by borrowing the money from her line of credit. How
much interest will she save? (Ans: $1,593.52)
Class work/HW: p.115 #3, 7, 10, 11
MATH 12 F – UNIT 2 LESSON 5 – BUY, RENT OR LEASE
Terminology:
Appreciation: An increase in the value of an asset over time. Usually, appreciation is stated as a yearly percentage. For
example, ‘The land appreciates 5% each year’.
Depreciation: A decrease in the value of an asset over time. Usually, depreciation is stated as a yearly percentage. For
example, ‘The car depreciates 15% each year’.
A comparison of buying, renting and leasing assets:
Description
BUY
- you own the asset
- you pay maintenance and
insurance
- you usually make a down
payment, then take out a loan
to pay the remainder of the
cost
Example 1:
Car/electronics/
equipment
- The asset usually depreciates
in value
- at the end of the asset’s life,
you can sell it for a small
amount
Example 2:
housing
- property usually appreciates
in value. For this reason, some
people consider the purchase of
a property as an ‘investment’.
- you pay a down payment,
then monthly mortgage
payments, monthly property
tax, utilities bills and property
insurance
- you pay for maintenance
- monthly mortgage payments
are usually much higher than
leasing a similar property
RENT
- you pay a fee to use
someone’s asset, usually for a
small period of time (usually a
day)
- you don’t pay for maintenance
- insurance is sometimes
included in the daily price
- you do not pay a down
payment
- for example, a daily rental of a
car
- you pay a daily fee, and
usually you pay an insurance
fee
- you do not pay for
maintenance costs
- one night rentals, for example,
a hotel room
- you pay a fee per night
- you do not pay for utilities,
maintenance or property tax
LEASE
- long term rental (usually at
least a year)
- you pay for some
maintenance
- You might pay for insurance
- sometimes you pay a down
payment
- for example, leasing a car
- you usually pay a down
payment, then a monthly fee
- you usually pay for some or all
maintenance costs and
insurance
- some leasing agreements have
a ‘lease to own’ option. For
example, you can lease a car for
4 years, after which you
automatically own the car
- sometimes referred to as
‘renting’
- usually, you sign a one year
lease
- you pay an initial ‘damage
deposit’ that you get back when
you move out, provided that
you didn’t damage anything
- you pay a monthly payment
and utilities
- you do not pay for
maintenance or property tax
Example 1: Compare the options of renting, leasing, or buying a car, if you need to use the car for approximately 12 days
each month.
-
-
You can rent for $49.99/day including tax (including insurance)
You can lease for $380/month including tax, for 4 years. The leasing agreement states that you will pay $4,000
down payment. You estimate that the maintenance costs will be about $50/month. Insurance will cost about
$1,220/year.
You can buy the car for $20,000 with a down payment of $3,000, financed over a 4 year term, with monthly
payments at 4.5% compounded monthly. You estimate that the maintenance costs will be about $60/month,
and insurance will cost about $1,220/year.
Calculate the cost of each option, and then decide which option is the best.
What other considerations should you use in making your decision?
Class work/HW: p.129 #1, 3, 4, 7
MATH 12 F – UNIT 2 LESSON 6 – APPRECIATION AND DEPRECIATION
Appreciation: An increase in the value of an asset over time. Usually, appreciation is stated as a yearly percentage. For
example, ‘The land appreciates 5% each year’.
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Depreciation: A decrease in the value of an asset over time. Usually, depreciation is stated as a yearly percentage. For
example, ‘The car depreciates 15% each year’.
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Example 1: Allen bought a property for $500,000. He estimates that the value of the property will appreciate 4% each
year.
a) Show the value of the property for each of the next 5 years.
b) Calculate the value of the property 20 years from now.
Example 2: A brand new car that cost $20,000 is expected to depreciate at a rate of 30% each year.
a) Calculate the value of the car for each of the next 5 years.
b) Calculate the value of the car 10 years from now.
Example 3: Lance needs an office space for his business. He is considering whether he should lease or buy.
-
He could sign a 3-year lease on an office space, with monthly rental payments of $1,000, and a refundable
damage deposit of $1,000.
He could purchase a house for $285,000 and renovate so it could be used as an office. A 5% down payment
would be required, and he would take out a 15-year mortgage at 5%, compounded semi-annually, with monthly
payments. The property is expected to appreciate in value 2% each year.
Compare the total costs over the next 15 years of leasing versus buying, and make a recommendation.
What other considerations should Lance use in making his decision?
Class work/HW: p. 130 #2, 5
UNIT 2 SUMMARY – FINANCIAL MATHEMATICS: BORROWING MONEY
UNIT OBJECTIVES:
-
-
Understanding the relationship between earning interest and paying interest when investing and borrowing
money
Using the simple interest formula = (1 + ), to calculate an unknown A, P, r or t
Using the compound interest formula = (1 + ) , to calculate an unknown A, P or i. (limitation of the
formula is that you cannot solve for an unknown n)
Using the annuity formula = ()
, to calculate an unknown PV or R. (limitations of the formula are:
you cannot solve for an unknown i or n; you can only use the formula if payments per period and compounds
per period are equal)
Using the TVM Solver app on the graphing calculator to solve compound interest and annuity loan problems
Understanding the difference between simple interest, compound interest and annuity, and being able to
identify from a word problem the type of investment the problem refers to.
Calculate the total interest paid for any type of loan
How mortgages work (downpayment and mortgage payments)
How credit cards work
Evaluate costs and benefits of a variety of credit card options (incentives, cash rebates, promotions, annual fees)
Evaluate the costs and benefits of buying, renting or leasing.