§8.4, Annuities, Stocks, and
Bonds
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1
Learning Targets
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.
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2
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.
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3
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
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4
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 = $3246.40
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
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5
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
t

P 1  r   1
.
A 
r
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6
Example 2: Determining the Value of an
Annuity
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?
b. Find the interest.
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7
Example 2: Determining the Value of an
Annuity continued
Solution:
a. The amount that you will have in the IRA is its value
t
P 1  r   1
after 30 years.


A
After 30 years, you will have
approximately $164,494 from the
IRA.
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r
30
1000 1  0.10   1



0.10
30
1000 1.10   1



0.10
1000 17.4494  1

0.10
1000 16.4494 

0.10
 164, 494
8
Example 2: Determining the Value of an
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.
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9
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


r
P 1    1
n


A
.
r
n
nt
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Example 3: Determining the Value of an
Annuity
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?
b. Find the interest.
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Example 3: Determining the Value of an
Annuity continued
Solution:
a. The amount that you will have in the IRA is its value
after 40 years.


r
nt
At age 65, you will have
approximately $604,765 from
IRA.
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P 1    1
n


A
r
n
1240


0.075 
200 1 

1


12





0.075
12
480
200 1.00625  1

0.00625
the
20019.8989  1

0.00625
 604,765


12
Example 3: Determining the Value of an
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.
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13
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
r
A 
n

P
.
nt


r
1    1
n


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Example 4: Achieving a Financial Goal
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?
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Example 4: Achieving a Financial Goal
continued
Solution:
a.
r
A 
n
P
nt


r
1    1
n


 0.06 
20, 000 

12 

P
125


0.06 
 1 
  1
12 


P  287
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You should deposit $287 each
month to be sure of having
$20,000 for a down payment
on a home in 5 years.
16
Example 4: Achieving a Financial Goal
continued
b. Total deposits = $287  12  5
= $17,220
Interest = $20,000  $17,220
= $2780
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17
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.
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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
20,000
 0.02  2%
1,000,000
• Any investor who owns some percentage of the
company is called a shareholder.
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19
Stocks
• 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.
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20
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.
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21
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.
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Reading Stock Tables
• 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.
Yesterday' s closing price per share
PE ratio 
Annual earnings per share
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Example 5: Reading Stock Tables
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?
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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
Yesterday' s closing price per share
Annual earnings per share 
PE ratio
52-week
High
Low
Stock
SYM
Div
42.38
22.50
Disney
DIS
.21
Yld
PE
%
.6
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43
Vol
100s
Hi
Lo
Close
Net
Chg
115900
32.50
31.25
32.50
…
25
Example 5: Reading Stock Tables
52-week
High
Low
Stock
SYM
Div
42.38
22.50
Disney
DIS
.21
Yld
PE
%
.6
43
Vol
100s
Hi
Lo
Close
Net
Chg
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.
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Example 5: Reading Stock Tables
52-week
High
Low
Stock
SYM
Div
42.38
22.50
Disney
DIS
.21
Yld
PE
%
.6
43
Vol
100s
Hi
Lo
Close
Net
Chg
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.
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Example 5: Reading Stock Tables
h. We use the formula given along with 43 for the PE ratio and
$32.50 for the closing price per share.
Yesterday' s closing price per share
PE ratio
$32.50

43
 $0.76
Annual earnings 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.
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Homework
Pg 478 – 480, #1 – 18, 23 – 32
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