Interest

Some people keep money at home in an easily accessible location, perhaps a piggy bank, a safe or locked box, or perhaps even a mattress. Although this provides instant access to funds it does not provide any return or earnings on this money. For that reason, most people hold their money in accounts or investments that provide some form of return or earning power.

Interest is the price paid for the use of money.  If you borrow money from another person or a lending institution, eventually you must pay back this amount plus the interest owing.  When you deposit money in a bank, you are lending them money and after some time they will pay you interest on the money you lent them.

The amount of interest you will owe or receive is determined by the principal,  the interest rate,  and the time (the length of the loan).  The amount of money that you lend or borrow is called the principal. The length of the loan can range between a few days to several years. The interest rate is stated as an annual percentage. It may be simple interest or compound interest. With simple interest the interest is calculated only once during the entire time period of the loan or deposit.  Simple interest is calculated solely on the principal investment or loan. With compound interest the interest is calculated more than once during the time period of the loan. It will be calculated on the principal as well as the accumulated interest This section will focus on simple interest and in the next section we will consider compound interest.

Simple Interest

Simple interest is calculated by finding the product of the principal (P), the rate (r), and the time (t).

Interest rates are quoted for periods of one year and when used in a formula must be converted to a decimal fraction. The time must be expressed in the same unit of time as the interest rate so time must be stated in years or portions of a year. If you deposit money in a savings account earning 3% interest then the annual interest rate is 3% per year.

EXAMPLE 1

Jo borrows $2000 at an interest rate of 5% per year.  How much interest will Jo owe after one year?

Solution

Identify the P, r, and t.     P = $2000     r = 5%  = 0.05      t = 1 year

Here,
			Replace P, r and t with their values
			Change 5% to its decimal equivalent, 0.05

Start with  the formula      I = Prt

I = 2000 (5%) (1)        Replace P, r, and t with their values

I = 2000 (0.05) (1)      Change 5% to its decimal equivalent, 0.05

I = 100

Jo will pay $100 in interest.

Note that the time t is expressed in terms of  years. When the time period is not exactly one year, the value for  t  will be the fraction of the year during which interest is earned.

If the investment is made for 3 months, then t = 3 months/12 months = 0.25 years.

If the investment is made for 35 days then  t = 35 days/365 days = 7/73 years.

EXAMPLE 3

Determine the interest that will be earned on  a deposit of $1350 at 2.8% over:

a) 7 months

b) 25 days

Solution:

a)

Interest of $22.05 over 7 months

b)

Interest of $2.59 over 25 days. Note that the answer is rounded to the nearest two decimal places or to the nearest cent.

EXAMPLE 4

Determine the interest that will be earned on  a deposit of $4200 at 4.65% over:

a) years

b) 5 weeks

Solution

a)

Interest of $292.95 over years

b)

Interest of $18.78 over 5 weeks. Note that the answer is rounded to the nearest two decimal places or to the nearest cent.

Maturity Value

The total amount of money due at the end of a loan period is called the maturity value of the loan. It is the amount to be paid on the due date of a loan or the amount to be paid to an investor at the end of the period for which an investment has been made.

In Example 1, Jo borrowed $2000 at an interest rate of 5%. At the end of one year Jo owed $100 in interest.

The maturity value of the loan is  MV = P + I  where  P = $2000 and I = $100.

MV = $2000 + $100 = $2100

The maturity value of the loan is $2100. At the end of the year Jo will be expected to pay back $2100.

EXAMPLE 5

Linda lends Ed $500.  Ed says he will pay her back in 60 days at 9% simple interest.  How much interest should Linda receive?  How much must Ed pay Linda altogether?

Solution

		Replace P, r and t with their values
		Multiply
		Round to the nearest cent

MV = P + I = $500 + $7.40 = $507.40

Linda should receive $7.40 in interest. At the end of 60 days Ed will owe Linda  $507.40.

Variations On Simple Interest

The amount of interest earned on an investment or due on a loan is calculated using I = Prt.

This formula can also be used to determine:

• the amount of principal (P) that needs to be invested in order to earn a certain amount of  interest over a certain period of time.
• the interest rate  (r)  that is needed in order to earn a certain amount of interest over a given time period.
• the amount of time (t) it will take in order to earn a certain amount of interest at a stated interest rate.

These amounts can be determined by solving the simple interest formula for any of  r, P or t.

Finding the Principal, Interest Rate, or Time

		where
To determine the principal use:		I = interest earned
To determine the interest rate use:		r = annual interest rate
To determine the time use:		P = principal
		t = time (in years)

The following memory aid is often called the “Magic Triangle”, because if you cover the variable you are trying to find, the formula will magically appear!

Determining the Principal

EXAMPLE  6

A six month investment will earn 5.25%.  How much would you need to invest if you want to earn $100 in interest?

Solution

The principal is unknown.  Cover P in the Magic Triangle.

P = ? or  appears. Use the formula:

or		Replace I, r and t with their respective values
		Multiply 0.0525 by 0.5
		Divide 100 by 0.02625 and round answer to nearest cent

You would need to invest $3809.52

Determining the Interest Rate

EXAMPLE 7

Mariko had $240 in the bank for the month of April.  At the end of the month she had earned $0.90 in interest.  What interest rate was the bank paying?

Solution

The interest rate is unknown.  Cover r in the Magic Triangle.

r = ? or appears. Use the formula:

t = 1 month =

Since r = 0.045, the interest rate as a percentage is 4.5%

Determining the Time

EXAMPLE 8

Carol invested $500 at an interest rate of  6%.  How long will it take her to earn $250 in interest?

Solution

The time is unknown.  Cover t in the Magic Triangle.

t = ? or appears. Use

r = 6% = 0.06
	or years

It will take 8.33 years.

When calculating time “t” using the simple interest formula, the answer will be in terms of years. Sometimes it is more reasonable to express the answer in terms of days or months.

Key Concepts

• to calculate the simple interest earned on an investment or charged on a loan we use the formula     I = Prt   where:

I = interest earned               r = annual interest rate ( stated as a decimal)

P = principal amount              t = time (in years)

• to determine the principal amount (P) for simple interest applications:
  • 
• to determine the time in years (t) for simple interest applications:
  • 
• to determine the interest rate (r) for simple interest applications:
  • 
• The Maturity Value (MV) of a loan is the sum of the principal P plus the interest I:

  MV = P + I

Glossary

Attribution

Some of the content for this chapter is from “Unit 1: Simple interest” and “Unit 2: Variations on simple interest” in Financial Mathematics by Paul Grinder, Velma McKay, Kim Moshenko, and Ada Sarsiat, which is under a CC BY 4.0 Licence.. Adapted by Kim Moshenko. See the Copyright page for more information.

9.2 Compound Interest

Compound Interest

We have seen that with simple interest an investment will earn interest on the original amount. For an investment of $100 earning 10% simple interest, the interest earned after one year will be $10 since 10% of $100 = $10. An investment will grow more quickly if the interest  is calculated more often than once a year.  Interest will not only be calculated on the principal amount but also on the previously earned interest.  This process is referred to compounding.

Figure 1 illustrates the process of compounding or earning interest on interest. Consider an investment of $100 that earns 10%/year with interest being compounded semiannually.  With semiannual compounding the interest on the investment will be calculated twice during the year.

Using the simple interest formula I = Prt,  at the end of six months (half a year) interest will be calculated as follows:

I = $100 x 10% x 1/2 year = $5.

Adding this $5 to the principal of $100 you will have $105 at the end of the first six months. At the end of the year interest will be calculated again on the $105:

I = $105 x 10% x 1/2 year = $5.25.

Adding this $5.25 to $105 you will have $110.25 at the end of the year. In this case you would be earning interest not only on the original principal of $100, but also on the previously earned interest of $5. When interest is earned on interest, we say the interest is compounded. The total amount of principal and accumulated interest at the end of a loan or investment is called the compound amount. 

Consider a $100 investment that earns 10%/year compounded annually. The table in Figure 2 shows how the value of the $100 investment will grow over a 6-year period.

Fig. 2

At the beginning of Year 1, $100 is invested, so the interest earned in the first year will be:

I = Prt = $100 × 0.10 × 1  = $10. This is added to the original $100 to result in $110 at the end of Year 1.

At the beginning of Year 2 the process will repeat but the principal P is now $110.

I = Prt = $110 × 0.10 ×1 = $11 in interest so at the end of Year 2 there will be:

$110 + $11 = $121 in the account.

Notice that the compound amount at the end of the six year period is $177.16.  The investment has earned an accumulated $77.16 in interest. If the investment had earned simple interest as opposed to compound interest it would have only earned:

I = Prt = 100 × 0.10 × 6 =  $60 in interest.

The above method of calculating the compound amount is very time consuming. Fortunately, there is a mathematical formula that we can use when working with compound interest.

Compound Interest Formula

The compound interest formula is:

	where,	A = total compound amount(includes principal and interest)
		P = principal
		r = annual interest rate
		n = number of times in one year that interest is calculated
		t = time (in years)

Since A includes both the principal and interest, to find the interest amount I calculate:

EXAMPLE 1

Find the compound amount and the interest earned on $100 compounded annually at 10% for 6 years.

Solution

P = $100 r = 10% = 0.1 n = 1 (since the interest is calculated once a year) t = 6 years		Replace the variables with their values
		and
		Raise

The interest earned is $77.16

The compound amount is $177.16

Interest can be compounded using a variety of compounding periods.   The compounding period is the span of time between when interest is calculated and when it will be calculated again. If there is one month between every interest calculation then the compounding period is monthly.  With monthly compounding there will be 12 compounding period in one year since there are twelve months in a year . The variable n in the compound interest formula reflects the number of times in one year that interest is calculated.

EXAMPLE 2

Find the compound amount and the interest earned on $500 compounded semiannually at 6% for 3 years.

Solution

(since the interest is calculated semiannually or 2 times a year)

The compound amount is $597.03 and the interest earned is $597.03 – $500 = $97.03

The greater the number of compounding periods in a year, the greater the total interest earned will be.

EXAMPLE 3

Find the compound amount and the interest earned on $500 compounded daily at 6% for 3 years.

Solution

(since the interest is calculated daily)

The compound amount is $598.60 and the interest earned is $598.60 – $500 = $98.60

Loan recipients must repay the principal amount borrowerd plus any interest charged. They will pay a greater price (in terms of total interest) when interest is  compounded.

EXAMPLE 4

Pat borrows $3200 an interest rate of  6.5% compounded semiannually. The original loan amount plus interest must be paid back in 3 years. Calculate the total amount that must be paid back in three years and determine the interest amount.

Solution

(since the interest is calculated semiannually or 2 times a year)

The compound amount is $3876.95 and the interest owing is $3876.95 – $3200 = $676.95

EXAMPLE 5

Pat borrows $3200 an interest rate of  6.5% compounded monthly. The original loan amount plus interest must be paid back in 3 years. Calculate the total amount that must be paid back in three years and determine the interest amount.

Solution

(since the interest is calculated monthly or 12 times a year)

The compound amount is $3886.95 and the interest owing is $3886.95 – $3200 = $686.95

Variation on Compound Interest – Present Value

We might want to know how much money we should invest now in order to make a purchase in the future.   Say for example that you want to know how much principal you needed to invest now in order to have $2000 in two years. The amount you need to invest now is called the present value of $2000. It is the amount of money that if invested now will accumulate to $2000 in two years. Assuming that your investment earns interest, the amount required now will be less than the future amount. Assuming annual compounding at an interest rate of 5% you will need to invest $1814.06 now to have $2000 in two years. Refer to Figure 3 below.

In Figure 3 we can see that at the beginning of the year $1814.06 is invested.

At the end of the first year the interest earned on $1814.06 is  5% × 1814.06 = $90.70.

At the end of the first year there will be $1904.76 in the account. In the second year the interest earned on $1904.76  is  $1904.76 × 0.05 = $95.24.

At th eend of the second year there will be $2000.00 in the account. The $2000 that is needed in two years is said to have a present value of $1814.06.

The present value can be calculated by solving the compound interest formula for P.

EXAMPLE 6

A house painting company is planning to expand its operations in three years time. It will require $24,000 in order to expand. How much must it invest now, at 4.6% interest compounded annually?

Solution

P = ? A = $24000 r = 4.6% = 0.046 n = 1
t = 3 years		Replace the variables with their values
		Add
		Raise

The present value is $20,970.86 so the company must invest that amount now to have $24,000 in three years.

EXAMPLE 7

Pat and her friends are planning a reunion in five years.  She estimates that the cost of the trip plus expenses will be approximately $2000.  How much should she invest right now in order to have $2000 five years from now, if she knows her money will earn 6% compounded quarterly?

Solution

P = ? A = $2000 r = 6% = 0.06 n = 4
t = 5 years		Replace the variables with their values
		Multiply
		Divide
		Add
		Raise

The present value is $1484.94

In other words, if Pat invested $1484.94 now at 6% compounded quarterly, then in 5 years the compound amount would be $2000.

Key Concepts

• to determine the compound amount (A)  of an investment or loan:
  • 
• to determine the interest component (I) of a principal or original amount (P) that has grown to a compound amount (A):
  • Interest = Compound Amount – Principal Amount    I = A – P
• to determine the present value P of a compound amount the formula is:
  • 
• Compounding Periods
  • annually  (once per year) ⇒ n = 1
  • semi-annually (twice a year)  ⇒ n = 2
  • quarterly (four times per year)  ⇒ n = 4
  • monthly  (twelve times per year)  ⇒ n = 12
  • weekly (fifty-two times per year)  ⇒ n = 52
  • daily  (three hundred sixty-five times per year) ⇒ n = 365

Glossary

Attribution

Some of the content for this chapter is from “Unit 3: Compound interest” and “Unit 4: Variations on compound interest – present value” in Financial Mathematics by Paul Grinder, Velma McKay, Kim Moshenko, and Ada Sarsiat, which is under a CC BY 4.0 Licence.. Adapted by Kim Moshenko. See the Copyright page for more information.

9.3 Interest Rates

Nominal and Effective Rates of Interest

As consumers and investors we are inundated with all kinds of offers for both investments and financing. When we require a loan we seek the lowest possible interest rate but  when we invest we want the highest rate of return. As we search for the best offer it is important to recognize that the advertised interest rate is not necessarily the true interest rate.

An advertised rate of 10% with annual compounding works out to be equivalent to a rate of 10% annually. We refer to this stated rate of 10% as the nominal interest rate.  An advertised rate of 10% with daily compounding works out to be equivalent to a rate of 10.52%  since interest is calculated on the interest 365 times in one year. We refer to the 10.52% as the effective interest rate.

The stated interest rate is referred to as the nominal interest rate, as it describes the named or numerical value. It is the rate that is usually stated in advertisements. The actual interest rate, or effective interest rate, reflects the real rate of return as it takes the compounding periods into account.

With simple interest calculations (where there is no compounding),  the stated annual interest rate indicates the true rate of return. If $1000 is borrowed at 6% for one year, then the interest owined will be  I = Prt = $1000 × 0.06/yr × 1yr = $60.

With compound interest calculations,  the stated annual interest rate does not indicate the true interest cost. If $1000 is invested at 6% for one year compounded semiannually, then the interest owed will be  $60.90 rather than $60.

One way to determine the  effective interest rate is to divide the total compound interest for the first year by the principal amount. If in the first year $60.90 is the interest charged on a principal of $1000 then the effective interest rate is  $60.90/$1000 = 0.0609 = 6.09%.  Although the nominal interest rate is 6%, the effective rate is 6.09%.

It is also possible to use a formula to calculate the effective interest rate.

Effective Interest Rate Formula

The effective interest rate formula is:

	where	f = effective interest rate
		r = nominal interest rate (annual interest rate)
		n = number of times in one year that interest is calculated

EXAMPLE 1

Use the formula to determine the effective interest  rate for 6% compounded annually.

Solution

f = ? r = 6% = 0.06 n = 1
		Replace the variables with their values
		Add

The effective interest rate is 6%. Note that the nominal and effective rate are the same since the number of compounding period is one (n = 1).

EXAMPLE 2

Use the formula to determine the effective interest  rate for 6% compounded monthly.

Solution

f = ? r = 6% = 0.06 n = 12
		Replace the variables with their values
		Divide
		Add
		Raise

The effective interest rate is 6.17%

EXAMPLE 3

What is the effective rate for a nominal rate of 9.8% compounded weekly?

Solution

f = ? r = 9.8% = 0.098 n = 52
		Replace the variables with their values
		Divide
		Add
		Raise

The effective interest rate is 10.29%

It is important to consider the effective interest rate, rather than the nominal rate,  when deciding on investments or loans.

Consider Bank A which offers a savings plan at 6.25% compounded monthly and Bank B which offers 6.5% compounded semi-annually.  Which of the two banks offers the better rate of return?  Although both banks offer the same nominal interest rate, their effective rates differ.  The effective rate will reflect the actual rate of return in one year. Example 4 will illustrate this.

EXAMPLE 4

Bank A offers 6.25% compounded monthly while Bank B offers 6.5% compounded semi-annually.  Which bank offers the better effective rate of return?

Solution

For Bank A:

f = ? r = 6.25% = 0.0625 n = 12

For Bank B:

f = ? r = 6.5% = 0.065 n = 2

Note that Bank A’s effective rate, 6.43%, is less than both Bank B’s nominal rate of 6.5% and Bank B’s effective rate of 6.61%. Bank B offers the better effective rate of return.

EXAMPLE 5

Consider two options for a 2 year loan. Bank A will charge 7.2% compounded monthly while Bank B will charge 7.4% compounded semi-annually.  Which bank offers the less expensive loan (charges the lower effective rate)?

Solution

For Bank A:

f = ? r = 7.2% = 0.072 n = 12

For Bank B:

f = ? r = 7.4% = 0.074 n = 2

Bank A’s effective rate, 7.44%, is less than Bank B’s effective rate of 7.54. By a slight margin, Bank A offers the less expensive loan.

Key Concepts

• to determine the effective annual interest rate (f):
  • 
• when investing money you want the higher effective interest rate; when borrowing money you want the lower effective interest rate.

Glossary

Attribution

Some of the content for this chapter is from “Unit 5: Nominal and effective rates of interest” in Financial Mathematics by Paul Grinder, Velma McKay, Kim Moshenko, and Ada Sarsiat, which is under a CC BY 4.0 Licence.. Adapted by Kim Moshenko. See the Copyright page for more information.

9.4 Loans and Mortgages

Installment Loans

A loan is something that is borrowed. In the case where this is a sum of money the amount that will be paid by the borrower will  include the original amount plus interest.

Some loans require full payment  on the maturity date of the loan.  The maturity date is when all principal and/or interest must be repaid to the the lender. Consider a one year loan of $1000  at a simple interest rate of 5%. At the end of one year (the maturity date) the borrower will pay back the original $1000 plus the interest of $50 for a total of $1050.

For major purchases such as vehicles or furniture there is a different type of loan, called the installment loan. The average consumer cannot afford to pay $25000 or more for a new vehicle and they  may not want to wait three or four years until they have saved enough money to do so. The qualifying consumer has the option of paying for the item with an installment loan.

Installment loans do not require full repayment of the loan on a specific date. With an installment loan the borrower is required to make regular (installment) payments until the loan is paid off. Each installment payment will include an interest charge. An installment loan can vary in length from a few years to perhaps twenty years or more (in the case of real estate).

Consider an installment loan for a $4000 television. The purchaser takes out a $4000 loan with a four-year term at an interest rate of 4.5%. The monthly installment payments will be $91.21. Although the television has a purchase price of $4000, the total cost to the purchaser will be more than $4000. The total of the installment payments will be:

Total Installment Payments = Number of Installment Payments x Payment Amount =

4 years x 12 payments/year x $91.21/mth  = $4378.08

The $4000 television ends up costing $4378.08 because the consumer is charged interest. Each payment includes an interest component that adds to the overall cost of the item. The total of the interest charges is referred to as the finance charge on the loan.

For the $4000 television the finance charge will be calculated as follows:

Finance charge = Total Installment Payments – Loan Amount =

(4 years x 12 payments/year x $91.21/payment) – $4000 = $4378.08 – $4000 = $378.08

Over the 4-year term of the loan the purchaser will have paid the $4000 loan amount plus an additional $378.08 in interest (the finance charge).

Sometimes the borrower will make an initial payment at the time of purchase. This is called a down payment. When a down payment is made the remaining amount is the amount financed or the  loan amount.

Consider the $4000 television. Assume the purchaser makes a down payment of $1500.

The amount financed is:  Purchase Price – Down Payment = $4000 – $1500 = $2500.

In this case the purchaser borrows $2500 rather than $4000. The amount financed is therefore $2500. Assuming the same 4-year term and an interest rate of 4.5%, the installment payments on the $2500 will be reduced to $57.01 per month. In this case the finance charge will be calculated as follows:

Finance charge = Total Installment Payments – Loan Amount =

(4 years x 12 payments/year x $57.01/payment) – $2500 = $2736.48 – $2500 = $236.48

With the down payment of $2500 the total finance charges will be reduced to $236.48 from $378.08.

The total cost of the television to the purchaser will be:

Purchase Price + Finance Charge

= $4000 + $236.48 = $4236.48

Alternatively we can calculate:

Total  Installment Payment + Down Payment

= $2736.48 + $1500 = $4236.48

As one can see, the finance charges are a hidden but added cost. This cost will become more pronounced with more expensive purchases such as with real estate.

Loan Payments

EXAMPLE 3

Refer back to the purchase of a television for $4000. The purchaser agrees to a 4 year term at an interest rate of 4.5%.   a) Use the formula to determine the monthly installment payment  b) Determine the total installment  payments

Solution

a)

where P = payment (unknown), A = $4000, r = 4.5%, n = 12, t = 4 years

 

The monthly payment is confirmed to be $91.21.

b) Total installment payments =  monthly payment amount x no. of payments

$91.21 x 48= $4378.08

TRY IT  3

A dining room table set is purchased for $5600. The purchase is financed  with a 3 year loan at an interest rate of 12.5%. a) Use the formula to determine the monthly installment payment  b) Determine the total installment  payments.

Show answer

Monthly payment is $187.34; Total Installment payments

EXAMPLE 4

Paul purchased a home entertainment system at a total cost of $6000. He obtained a 3 year loan at an interest rate of  7.5%.  Use the formula to determine his monthly payments. Confirm that this matches the amount in Example 1.

Solution

		P = payment (unknown)
	where	A = $6000          r = 7.5%
		n = 12                 t = 3 years

 

The monthly payment is confirmed to be $186.64

TRY IT 4

Cassie purchased a new washer and dryer at a total cost of $3800. She obtained a 4 year loan at an interest rate of  6.2%.  Use the formula to determine her monthly payments. Confirm that this matches the amount in Try It 1.

Show answer

Monthly payment of $89.59 is confirmed

EXAMPLE 5

Determine   a) the annual payments   b)the total installment payments  and  c) the finance charge on a 5 year loan of $5000 where payments are made annually and the interest rate is 6%.

Solution

a)

P = payment

A = $5000    r = 6%

       n = 1            t = 5 years

The annual payment will be $1186.98.

b) Total  installment payments = $1186.98 x 5 = $5934.90

c) Finance charge = $5934.90 – $5000 = $934.90

TRY IT 5

Determine a) the annual payments   b) the  total installment payments  and  c) the finance charge  on a 5 year loan of $5000 where payments are made monthly and the interest rate is 6%.

Show answer

a) Annual payment is $96.67

b) Total Installment payments = $5800.20

c) Finance charge $800.20

Recall that interest is calculated only on the loan amount and not on any downpayment. When determining the periodic payment on an installment loan be sure to exclude the downpayment when  calculating the periodic payment.

EXAMPLE 6

Mike purchased a home entertainment system at a total cost of $6000. He made a down payment of $1800 and to pay the balance he obtained a 3 year loan at an interest rate of  7.5%.   Use the formula to determine his monthly payments. Confirm that this matches the amount provided in Example 2.

Solution

		P = payment (unknown)
	where	A = $4200          r = 7.5%
		n = 12                 t = 3 years

 

The monthly payment is confirmed to be $130.65

TRY IT 6

Carl purchased a new washer and dryer at a total cost of $3800. He made a down payment of $1500 and obtained a 2 year loan for the remaining amount at an interest rate of  6.2%.  Use the formula to determine his monthly payments. Confirm that this matches the amount provided in Try It 2.

Show answer

Monthly payment of $102.14 is confirmed

EXAMPLE 7

Pat has decided to purchase a used vehicle that costs $12,500. He considers two options. For each option, determine a) the monthly payment  b) total installment payments  c) the finance charge for each option. What is the difference in the finance charge with the down payment?

Option 1) Paying the full amount with a 4 year loan, monthly payments, and an interest rate of 6.8%.

Option 2) He will cancel a planned trip and and instead make a  $3500 down payment on the purchase. He will pay the remaining amount with a 4 year loan, monthly payments, and an interest rate of 6.8%.

Solution

Option 1)

a) P = unknown      A = $12,500

r = 0.068        n = 12          t = 4

	payment

b) Total Installment payments

c) Finance charge = Total Installment Payments – Loan Amount = $14312.16 – $12,500 = $1812.16

Option 2)

a) P = unknown      A = $12,500 – $3500 =$9000

r = 0.068        n = 12          t = 4

	payment

b) Total Installment payments

c) Finance charge = Total Installment Payments – Loan Amount = $10,304.64  – $9000 = $1304.64

With a down payment there will be a savings of $507.52 on the finance charges.

TRY IT 7

Mick has decided to purchase a home entertainment system at a cost of  $9200. He considers two options. For each option determine a) the monthly payment  b) total installment payments c) the finance charge for each option. What is the difference in the finance charge with the down payment?

1) Paying the full amount with a 3 year loan that offers an interest rate of 8.4%.

2) Forgoing the purchase of a new electric  bike and instead makinga $2000 down payment on the bike purchase. He will pay the remaining amount with a 3 year loan at an interest rate of 8.4%.

Show answer

With no down payment: a) $290   b)  $10440   c) $1239.83

With a down payment  a) $226.95  b)   $10170.20   c) $970.30;  With the down payment the finance charge is $269.53 less

Amortization 

Key Concepts

• For an Installment Loan:
  • to determine the total installment payments:

    Number of Payments x Payment Amount

  • to determine the finance (interest) charge:

     Total Installment Payments  – Loan Amount

  • to determine the amount financed:

 Purchase Price – Down Payment

• • to determine the total amount paid for the item:

Purchase Price + Finance Charge

or

Total  Installment Payments + Down Payment

• • to determine the periodic payment P:

Glossary