CHAPTER 8gphelps.faculty.unlv.edu/120/bztm5e_ppt_8_4.pdf · © 2010 Pearson Prentice Hall. All rights reserved. 2 8.4 Annuities, Stocks, and Bonds

© 2010 Pearson Prentice Hall. All rights reserved.

CHAPTER 8

Consumer Mathematics

and Financial

Management

© 2010 Pearson Prentice Hall. All rights reserved. 2

8.4

Annuities, Stocks, and Bonds


Objectives

1. Determine the value of an annuity.

2. Determine regular annuity payments needed to

achieve a financial goal.

3. Understand stocks and bonds as investments.

4. Read stock tables.


Annuities

• An annuity is a sequence of equal payments made at

equal time periods.

• The value of an annuity is the sum of all deposits

plus all interest paid.


Example 1: Determining the Value of an

Annuity

You deposit $1000 into a savings plan at the end of each year for

three years. The interest rate is 8% per year compounded

annually.

a. Find the value of the annuity after three years.

b. Find the interest.

Solution:

a. The value of the annuity after three years is the sum of all

deposits made plus all interest paid over three years.

The value at the end of year 1 = $1000

The value at the end of year 2 = $1000(1 + 0.08) + $1000

= $1080 + $1000 = $2080



Annuity continued

The value at the end of year 3 = $2080(1 + 0.08) + $1000

= $2246.40 + $1000 = $3246.40

The value of the annuity at the end of three years is $3246.40.

b. You made three payments of $1000 each, depositing a total

of $3000.

The value of the annuity is $3246.40, the interest is

$3246.40 $3000 = $246.40


Value of an Annuity: Interest Compounded

Once a Year

If P is the deposit made at the end of each year for an

annuity that pays an annual interest rate r (in decimal

form) compounded once a year, the value, A, of the

annuity after t years is

1 1.

tP r

Ar


To save for retirement, you decide to deposit $1000 into

an IRA at the end of each year for the next 30 years. If

you can count on an interest rate of 10% per year

compounded annually,

a. How much will you have from the IRA after 30

years?



Annuity


Solution:

a. The amount that you will have in the IRA is its value

after 30 years.

After 30 years, you will have

approximately $164,494 from the

IRA.

30

30

1 1

1000 1 0.10 1

0.10

1000 1.10 1

0.10

1000 17.4494 1

0.10

1000 16.4494

0.10

164,494

tP r

Ar


Annuity continued


b. You made 30 payments of $1000 each, a total of

$30,000. Because the annuity is $164,494, the

interest is

$164,494 $30,000 = $134,494.

The interest is nearly 4½ times the amount of your

payments.


Annuity continued


Annuity

Interest Compounded N Times Per Year

If P is the deposit made at the end of each compounding

period for an annuity that pays an annual interest rate r

(in decimal form) compounded n times per year, the

value, A, of the annuity after t years is

.

11

n

r

n

rP

A

nt


At age 25, to save for retirement, you decide to deposit

$200 into an IRA at the end of each month at an interest

rate of 7.5% per year compounded monthly,

a. How much will you have from the IRA when you

retire at age 65?



Annuity


Solution:

a. The amount that you will have in the IRA is its value

after 40 years.

At age 65, you will have

approximately $604,765 from the

IRA.

765,604

00625.0

18989.19200

00625.0

100625.1200

12

075.0

112

075.01200

11

480

4012

n

r

n

rP

A

nt


Annuity continued



Annuity continued

b. Interest = Value of the IRA – Total deposits

≈ $604,765 $200 12 40

= $604,765 $96,000

= $508,765

The interest is approximately $508,765, more than five

times the amount of your contributions.


Planning for the Future with an Annuity

The deposit, P, that must be made at the end of each

compounding period into an annuity that pays an annual

interest rate r (in decimal form) compounded n times

per year in order to achieve a value of A dollars after t

years is

.

11

nt

n

r

n

rA

P


You would like to have $20,000 to use as a down

payment for a home in five years by making regular,

end-of-the-month deposits in an annuity that pays 6%

compounded monthly.

a. How much should you deposit each month?

b. How much of the $20,000 down payment comes

from deposits and how much comes from interest?

Example 4: Achieving a Financial Goal


Solution:

a.

12 5

1 1

0.0620,000

12

0.061 1

12

287

nt

rA

nP

r

n

P

P


continued

You should deposit $287 each

month to be sure of having

$20,000 for a down payment

on a home in 5 years.


b. Total deposits = $287 12 5

= $17,220

Interest = $20,000 $17,220

= $2780


continued


Investments

• When depositing money into a bank account, you are

making a cash investment.

• The account’s interest rate guarantees a certain

percent increase in your investment, called its return.

• Other kind of investments that are riskier are called

stocks and bonds.


Stocks

• Investors purchase stock, shares of ownership in a

company.

For example, if a company has issued a total of 1

million shares and an investor owns 20,000 of these

shares, that investor owns

• Any investor who owns some percentage of the

company is called a shareholder.

%202.0000,000,1

000,20


• Buying or selling stocks is referred to as trading.

• Stocks are traded on a stock exchange.

There are two ways to make money by investing in

stock:

• You sell shares for more money than you paid for

them, in which case you have a capital gain on the

sale of stock.

• While you own the stock, the company distributes all

or part of its profits to shareholders as dividends.

Stocks


Bonds

• In order to raise money and dilute the ownership of current

stockholders, companies sell bonds.

• People who buy a bond are lending money to the company

from which they buy the bond.

• Bonds are a commitment from a company to pay the price an

investor pays for the bond at the time it was purchased, called

the face value, along with interest payments at a given rate.

• A listing of all the investments that a person holds is called a

financial portfolio.

• Most financial advisors recommend a portfolio with a mixture

of low-risk and high-risk investments, called a diversified

portfolio.


Reading Stock Tables

Daily newspapers and online services give current stock

prices and other information about stocks.

• 52-week high refers to the highest price at which a

company traded during the past 52 weeks.

• 52-week low refers to the lowest price at which a

company traded during the past 52 weeks.

• Stock refers to the company name.

• SYM refers to the symbol the company uses for trading.

• Div refers to dividends paid per share to stockholders last

year.

• Yld% stands for percent yield.

• Vol 100s stands for sales volume in hundreds.


• Hi stands for the highest price at which the

company’s stock traded yesterday.

• Low stands for the lowest price at which the

company’s stock traded yesterday.

• Close stands for the price at which shares traded

when the stock exchange closed yesterday.

• Net Chg stands for net change.

• PE stands for the price-to-earnings ratio.

Reading Stock Tables

shareper earnings Annual

shareper price closing sYesterday' ratio PE


Use the stock table for Disney to answer the following questions:

a. What were the high and low prices for the past 52 weeks?

b. If you owned 3000 shares of Disney stock last year, what dividend did you receive?

c. What is the annual return for dividends alone? How does this compare to a bank account offering a 3.5% interest rate?

d. How many shares of Disney were traded yesterday?

e. What were the high and low prices for Disney shares yesterday?

Example 5: Reading Stock Tables


f. What was the price at which Disney shares traded

when the stock exchange closed yesterday?

g. What does “…” in the net change column mean?

h. Compute Disney’s annual earnings per share using

ratio PE

shareper price closing sYesterday' shareper earnings Annual

52-week

High Low Stock SYM Div

Yld

% PE

Vol

100s Hi Lo Close

Net

Chg

42.38 22.50 Disney DIS .21 .6 43 115900 32.50 31.25 32.50 …



Solution:

a. We look under the heading High and Low. The 52-week high

was 42.38 or $42.38. The 52-week low was 22.50 or $22.50.

b. We look under the heading Div for the dividend paid for a

share of Disney stock. So, the annual return of a share last

year was $0.21. If you owned 3000 shares, then you received

a dividend of 3000 x 0.21 = $630.

c. The heading Yld% is the percentage that give investors an

annual return of their dividends. For Disney, it is 0.6%. This

is much lower than a bank account paying 3.5% interest rate.


52-week


Yld

% PE

Vol

100s Hi Lo Close

Net

Chg

42.38 22.50 Disney DIS .21 .6 43 115900 32.50 31.25 32.50 …


d. We look under Vol100s for the number of shares Disney

traded yesterday. So, Disney traded 115900 x 100 =

11,590,000 shares yesterday.

e. The high and low prices for Disney shares yesterday are

found by looking under Hi and Lo. Hence, the high price was

$32.50 and the low price was $31.25 per share yesterday.

f. When the stock exchange closed yesterday Disney’s shares

were $32.50 per share.

g. The “…” under Net Chg means that there was no change in

price in Disney stock from the market close two days ago to

yesterday’s market close.

52-week


Yld

% PE

Vol

100s Hi Lo Close

Net

Chg

42.38 22.50 Disney DIS .21 .6 43 115900 32.50 31.25 32.50 …



h. We use the formula given along with 43 for the PE ratio and

$32.50 for the closing price per share.

Thus, the annual earnings per share for Disney are $0.76.

• The PE ratio tells us that yesterdays closing price is 43 times

the earnings per share.

76.0$

43

$32.50

ratio PE

shareper price closing sYesterday' shareper earnings Annual


CHAPTER 8gphelps.faculty.unlv.edu/120/bztm5e_ppt_8_4.pdf · © 2010 Pearson Prentice Hall. All rights reserved. 2 8.4 Annuities, Stocks, and Bonds

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