chapter 8

© 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


Example 1: Determining the Value of an Annuity continued

The value at the end of year 3 = $2080(1 + 0.08) + $1000= $2246.40 + $1000 =

$3246.40The 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?




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

0.101000 17.4494 1

0.10

1000 16.4494

0.10 164,494

tP r

Ar



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.



AnnuityInterest 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

nr

nr

P

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?




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

12075.0

112075.0

1200

11

480

4012

nr

nr

P

A

nt




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

nr

nr

AP


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

nPrn

P

P

Example 4: Achieving a Financial Goal 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

Example 4: Achieving a Financial Goal 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 DivYld %

PEVol

100sHi 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 High


PEVol

100sHi 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 High


PEVol

100sHi 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


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