UNIT 7 – Annuities

Date Lesson Text TOPIC Homework

Dec. 7 7.1

7.1

The Amount of an Annuity

with technology

Pg. 415 # 1 – 3, 5 – 7, 12

**check answers withTI-83

Dec. 9 7.2

7.2

The Present Value of an Annuity

with technology

Pg. 423 # 1 – 3, 5 – 8, 12

** 5 needs TI-83

Dec.

12

7.3

7.3

The Regular Payment of an Annuity

with technology

Pg. 430 # 2 – 4, 6 – 8, 11, 16

**check answers withTI-83

Dec.

13

7.4

7.6

What is a Mortgage

with technology

Pg. 445 # 1 – 5

**check answers withTI-83

Dec.

14

7.5

7.6

Amortizing a Mortgage

with technology

QUIZ (7.1 – 7.3)

Pg. 450 # 1 – 5, 8, 10, 13

**check answers withTI-83

Dec.

15

7.6

Re-Cap of Annuities

with technology

Pg. 442 # 1 – 3, 5, 6, 7b, 8b

OR WS 7.6

Use formulas only but check

with TI-83

Dec.

16

OPT

7.7

Amortizing a Mortgage II

with technology

ANNUITIES TECHNOLOGY QUIZ

In class with TI-83 if needed

Pg. 461 # 1 - 4

Dec.

19

7.8

Review for Unit 7 Test

Pg. 468 # 1 – 9, 11, 16, 17, 18

Dec.

21

7.9

TEST UNIT 7

MAP 4C Lesson 7.1 The Amount of an Annuity

Ordinary Simple An annuity is a series of equal payments made at regular intervals. In an

Annuities ordinary simple annuity, payments are made at the end of each compounding

period. The amount of an annuity is the sum of the regular payments plus

interest.

Ex. 1 Using a Table Suppose $450 is deposited at the end of each quarter for 1.5 years in an investment

account that earns 10% per year compounded quarterly.

Complete the following table to determine the amount of the annuity?

There has to be a quicker way.

Quarter Starting

Balance

Interest

Earned (2.5%) Deposit

Ending

Balance

1 $0.00 $450.00 $450.00

2 $450.00 $450.00 x 0.025

= $11.25 $450.00 $911.25

3 $911.25

$450.00

4 $450.00

5

$450.00

6

$450.00

Total

There is !

The amount formula can only be used when:

The payment interval is the same as the compounding period.

A payment is made at the end of each compounding period.

The first payment is made at end of the first compounding period.

Ex. 2 In the annuity in Ex. 1, $450 is deposited at the end of each quarter for 1.5 years at 10% per year

compounded quarterly.

a) Use the formula to determine the amount of the annuity?

iiR

FVn ]1)1[(

If you have a direct entry calculator,

ENTER is the same as =.

b) How much interest does it earn?

The regular payment is $450, so R = 450.

025.04

10.0i ; n = 1.5 x 4 = 6

Substitute R = 450, i = 0.025 and n = 6 into the amount formula.

a.k.a. Future Value (FV)

Ex. 2 In the annuity in Ex. 1, $450 is deposited at the end of each quarter for 1.5 years at 10% per year

compounded quarterly.

a) Use the TVM solver to determine the amount of the annuity?

If using the TI-83, press [2nd] [x-1] [1]

b) How much interest does it earn?

Annuities and Annuities are often used to save money for expenses such as a car, a down payment

regular savings for a house, or a vacation. They are also used to save for education and retirement.

Relatively small, regular deposits can accumulate to large sums of money over time.

Ex. 4 Amira and Bethany are twins. They save for retirement as shown below.

Starting at age 25, Amira deposits $1000 at the end of each year for 40 years.

Starting at age 40, Bethany deposits $2000 at the end of each year for 25 years.

Suppose that each annuity earns 8% per year compounded annually. Who will have the greater

amount at retirement?

You could use the formula or the TVM Solver to solve this problem. You must be able to use both.

Amira Bethany

Ex. 4 illustrates the power of time on the value of money and the advantage of starting to save early.

Pg. 415 # 1 – 3, 5 – 7, 12

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

MAP 4C Lesson 7.2 The Present Value of an Annuity

Present Value of The present value of an annuity is the principal that must be invested today

an Annuity to provide the regular payments of an annuity.

The present value formula can only be used when:

The payment interval is the same as the compounding period.

A payment is made at the end of each compounding period.

The first payment is made at end of the first compounding period.

Providing for an Annuity

Ex. 1 Hudson wants to withdraw $700 at the end of each month for 8 months, starting 1 month from now.

His bank account earns 5.4% per year compounded monthly. How much must he deposit in his bank

account today to pay for the withdrawals?

Method 1 – Use the PV formula

i

iRPV

n])1(1[ If you have a direct entry calculator,

The regular payment is $700, so R = 700.

0045.012

054.0i ; n = 8

Substitute R = 700, i = 0.0045 and n = 8 into the present value formula.

Method 2 – Use the TVM Solver

Ex. 2 Andrea plans to retire at age 55. She would like to have enough money saved in her account so she

can withdraw $7500 every 3 months for 30 years, starting 3 months after she retires. How much

must she deposit at retirement at 9% per year compounded quarterly to provide for the annuity?

The amount she must deposit is the present value of the annuity.

Method 1: Formula Method 2: TVM Solver

i

iRPV

n])1(1[

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

Repaying Loans Most loans are repaid by making equal monthly payments over a fixed period of time.

These payments form an annuity whose present value is the principal borrowed.

When all of the payments are made, both the principal borrowed and the interest

due will have been paid.

Ex. 1 Allison plans to buy a car. She can afford monthly payments of $300. The car dealer offers her a

loan at 6.9% per year compounded monthly, for 4 years. The first payment will be made 1 month

from when she buys the car.

a) How much can she afford to borrow?

The amount she can borrow is the (You could use the formula or the TVM Solver.)

present value of the loan.

b) How much interest will she pay?

Pg. 423 # 1 – 3, 5 – 8, 12

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

Rearranged Future Value Formula

11

ni

iAR

Rearranged Present Value Formula

n

i

iPVR11

MAP 4C Lesson 7.3 The Regular Payment of an Annuity

When we know the amount or the present value of annuity, we can solve for the regular payment.

To do this, we rearrange the appropriate formula to solve for R. We can do this before substituting

for all the known values or after substituting.

Ex. 1 Habeeba wants to save $3000 for the Japan trip in 3 years. What regular deposit should she make at

the end of every 6 months in an account that earns 6% per year compounded semi-annually?

Ex. 2 David borrows $1200 from an electronics store to buy a computer. He will repay the loan in equal

monthly payments over 3 years, starting 1 month from now. He is charged 12.5% per year compounded

monthly. How much is his monthly payment?

Ex. 3 Pershang borrows $9500 to buy a car. She has 2 options to repay the loan.

Option A: 36 monthly payments at 6.9% per year compounded monthly

Option B: 60 monthly payments at 8.4% per year compounded monthly

a) Use the TVM Solver to determine her monthly payment in each case?

Option A Option B

b) How much interest will she pay for each option?

Option A Option B

c) Explain why Pershang might choose each option.

Option A

Option B

Pg. 430 # 2 – 4, 6 – 8, 11, 16

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

MAP 4C Lesson 7.4 What is a Mortgage?

What is a mortgage?

A mortgage is a lien on a property/house that secures a loan and is paid in installments over a set period of time.

The mortgage secures your promise that you'll repay the money you've borrowed to buy your home.

What financial requirements must be met to qualify for a mortgage?

Must have employment, and your mortgage payments, interest and property tax must not be more than 32% of your

gross pay, and your payments, interest and property tax, heating and other monthly debt cannot exceed 40%.

What is the minimum down payment required for a mortgage?

Typically, a minimum down payment of 25% of the total loan balance is required to qualify for a loan without having

to pay private mortgage insurance.

How often is the interest compounded?

Canadian law only allows mortgages to be compounded semi-annually.

What is the difference between the amortization period and term of a mortgage?

The amortization period is the length over which the total cost of the mortgage will be paid back. The term of a

mortgage is shorter, and it is a period of time over which a certain interest rate will be paid on the mortgage. The

interest rate must be renegotiated at the end of each term in order to begin the next term.

What amortization periods and terms are commonly available?

Typical mortgage was amortized over 25 years. Shorter amortizations do exist, such as 10, 15, and 20 years. Yet

longer amortizations of 30, 35, 40 and 50 years are becoming more and more common. Mortgage terms can be

from 1 year up to 25 years, typically.

How often can mortgage payments be made?

Monthly, semi-monthly, bi-weekly and weekly. You can also make “accelerated” payments. Accelerated payments are

exactly half of a monthly payment amount, collected every two weeks. For example if the monthly payment is

$1,000 then the bi-weekly payment will be $500. This saves you money because you pay an extra $1,000 over a

twelve month period. If you pay $1,000 per month x 12 months = $12,000 in payments for the year, but if you pay

bi-weekly then it is $500 X 26 = $13,000.

The $1,000 a month payment is multiplied by 12, then divided by 26. This equals a bi-weekly payment of $461.54 -

at the end of the year you will have paid $12,000

Conventional mortgage or a high-ratio mortgage?

Conventional mortgage - If you have at least 25% of the purchase price (or appraised value if this is lower

than the purchase price) as a down payment, you can apply for a conventional

mortgage. Does not normally need to be insured.

High-ratio mortgage - If you have between 5% and 25% of the purchase price as your down payment, you can

apply for a high-ratio mortgage. Usually these have to be insured through CMHC

(Canada Mortgage and Housing Corporation)

Open or closed mortgage?

An open mortgage allows you to pay off part or the entire mortgage at any time without penalties. Open mortgages

usually have short terms of six months or one year. The interest rates are higher than those for closed mortgages

with similar terms. A closed mortgage cannot be paid off at any time without penalty.

Fixed-rate or variable-rate mortgage?

For a fixed rate, the interest rate is locked in for the term of the mortgage. For a variable rate mortgage, the

interest rate is not locked in and will change in accordance with the change of the PRIME interest rate. It is

possible the interest rate will rise over the course of the mortgage.

Mortgage interest Under Canadian law, interest on mortgages can be compounded at most semi-annually.

rates However, mortgage payments are often made monthly. These monthly payments form

an annuity whose present value is the principal originally borrowed.

Since the payment and compounding period are different, we cannot calculate the monthly

payment by using the formula for the present value of a n ordinary simple annuity. We use

the TVM Solver instead. To represent monthly payments and semi-annual compounding,

we set P/Y = 12 and C/Y = 2.

Ex. 1 The Smiths take out a mortgage of $210 000 at 5% per year compounded semi-annually

for 25 years.

a) What is their monthly payment?

ii) If they choose to make a 15% down payment,

how much will they need?

b) What is the total interest paid over the 25 years?

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

Ex. 2 Laura and Charles have to pay a 1.25% land-transfer tax on the home they bought.

If the purchase price of the house was $195 000. How much is the tax?

Ex. 3 Liban buys a $150 000 condo and decides to make bi-weekly payments. The bank offers a 20 year

mortgage at 3.4%/a compounded semi-annually. What is his bi-weekly payment?

Ex. 4 Solomon buys a $210 000 cottage and makes a 20% down payment. The bank offers a 25 year

mortgage at 4.1%/a compounded semi-annually. What is his monthly mortgage payment?

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

N =

I% =

PV =

PMT =

FV =

P/Y =

C/Y =

PMT: END BEGIN

Pg. 445 # 1 – 5 Use TI-83 for # 5b

MAP 4C Lesson 7.5 Amortizing a Mortgage

Amortizing a A mortgage is amortized when both the principal and the interest are paid off with a series

mortgage of equal, regular payments. For example, in the first example, the mortgage was amortized

by making monthly payments of $1221.37 over an amortization period of 25 years. To

simplify the math, we assumed that the interest rate was fixed for the entire amortization

period. In reality, mortgage interest rates are only fixed for a shorter length of time

called the term of the mortgage. The term normally ranges from 6 months to 10 years.

At the end of the term, the mortgage must be paid off or renewed at the current rate of

interest.

Amortization Table We can use an amortization table to analyse how a mortgage is repaid. The

amortization table gives a detailed breakdown of the interest and principal

paid by each payment and the loan balance after the payment.

This table shows the amortization of a small loan.

How much interest is paid over the term of the loan?

What is the total amount paid over the period of the loan?

Ex. 2 Below is a partial amortization table for the Smith’s mortgage.

a) How much interest and principal is paid in the 5th payment?

b) How much do the Smiths still owe after this payment?

c) What is the outstanding balance after 6 months?

d) Compare the interest and principal paid in the 1st 6 months of the mortgage with the interest and

principal paid in the last 6 months of the mortgage. What do you notice?

e) Why is the monthly payment increased for the 300th payment?

f) What percent of the total amount paid is interest?

Pg. 450 # 1 – 5, 8, 10, 13

OR

Pg. 442 # 1 – 3, 5, 6, 7b, 8b

WS 7.6

MAP 4C Lesson 7.7 Amortizing a Mortgage II

Ex. Shivika buys a condo for $200000. She makes a 10% down payment and finances the rest at 4.9%/a

compounded semi-annually amortized over 20 years. She has to decide whether to make monthly or

bi-weekly payments. What would you advise her to do? Justify your answer.

Monthly Bi-weekly Accelerated bi-weekly payments are half the monthly payment, made every two weeks (26 times per year).

Pg. 461 # 1 – 4 USE TI-83

Annuities Study Guide

How to use the TVM solver to determine the amount of the annuity

Calculating Mean, Median and Standard Deviation for a Sample with TI-83 Step 1 :

Press [STAT] [ 1 ] for STAT Edit

Enter the data into one of the lists Step 2:

Press [STAT]

Press right arrow key [] to highlight CALC

Press [ENTER] or [1] for 1-Var Stats

Enter the name of the list containing your data

Press [2nd ] [1] for L1, [2nd] [2] for L2 ....

Press [ENTER] On TI-83 screen

x = … (Sample mean)

Med = ... (Sample median)

x= … (Sample standard deviation)

MAP 4C Lesson (OPTIONAL) Creating an Amortization Table

An Amortization Table is a payment schedule that shows the amount that goes towards principle and interest of

a loan, and the balance owed after n payments.

Ex: Find the payment and the amortization schedule of a 30 year mortgage at $150,000 with an interest rate

of 8% per year compounded semi-annually using a TI-83.

Calculating the Payment:

• Press [APPS] and select Finance by pressing [ENTER]

• Select TVM_Solver by pressing [ENTER]

• Enter in the respective values for this mortgage N = 360, I% = 8, PV = 150,000, FV = 0, P/Y = 12, C/Y = 2

• Compute the monthly payment, place the cursor to the PMT = selection and press [ALPHA] [ENTER]

• The payment is equal to -1087.07

Now set up the Functions

• Go to the Y= editor by pressing [Y=]

• Y1 = bal(x) To enter this press [APPS] [ENTER] [9] [X,T,θ,n] [ ) ] [ENTER]

• Y2 = ∑Prn(X,X) To enter this press [APPS] [ENTER] [0] [X,T,θ,n] [ , ] [X,T,θ,n] [ ) ] [ENTER]

• Y3 = ∑Int(X,X) To enter this press [APPS] [ENTER] [ALPHA] [MATH] [X,T,θ,n] [ , ] [X,T,θ,n] [ ) ] [ENTER]

Now adjust the table set:

• Press [2nd] followed by [WINDOW]

• TblStart = 0

• ∆Tbl = 1

• Indpnt: AUTO

• Depend: AUTO

N = 360 I% = 8 PV = 150000 PMT = -1087.07 FV = 0 P/Y = 12 C/Y = 2 PMT: END BEGIN

Now view the table:

• [2nd] [GRAPH]

The X column is the payment number

Y1 represents the balance after payment X

Y2, is the amount of payment X that went towards principle

Y3 is the amount of payment X that went towards interest.

To view Y3, scroll to the right.

Pg. 460 # 4 - 6