Download 11.1 What You Will Learn

11.1 What You Will Learn
•
Percent, Fractions, and Decimal Numbers
(skip, assume that you known)
•
Percent Change
•
Percent Markup and Markdown
11.1-1
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Percent Change
Formula:
Percent change =
•
•
11.1-2
 amount in 
 amount in

 latest period −  previous period
amount in previous period
× 100
If percent change is positive, one gets % increase;
If percent change is negative, one gets % decrease.
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Example 6: Most Improved
Baseball Team
In 2009, the Padres won 75 games. In
2010, the Padres won 90 games. Determine
the percent increase in the number of
games won by the Padres from 2009 to
2010.
20% increase
11.1-3
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Percent Markup
percent markup or markdown on cost.
•
•
A positive answer indicates a markup.
A negative answer indicates a markdown.
Percent
markup or = selling price − dealer's cost × 100
dealer's cost
markdown
on cost
11.1-4
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Example 8: Determining Percent
Markup
Holdren Hardware stores pay $48.76 for
glass fireplace screens. They regularly sell
them for $79.88. At a sale they sell them for
$69.99. Determine
a)
the percent markup on the regular price.
63.8% increase
b) the percent markup on the sale price.
43.5% increase
11.1-5
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Example 10: Down Payment on a
Condominium Home
Melissa Bell wishes to buy a
condominium home for $189,000. To
obtain the mortgage loan, she must
pay 20% of the selling price as a down
payment. Determine the amount of
Melissa’s down payment.
Down payment: $37,800
11.1-6
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11.2 What You Will Learn
Ordinary Interest
The United States Rule (skipped)
Banker’s Rule (skipped)
11.2-7
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Simple Interest Formula
Interest = Principal × rate × time
i=P⋅r⋅t
11.2-8
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Example 1: Air Conditioning Loan
Sherry Tornwall needs to borrow $6200 to replace
the air conditioner in her home. From her credit
union, Sherry obtains a 30-month loan with an
annual simple interest rate of 5.75%.
a)Calculate
the loan.
the simple interest she is charged on
The simple interest is $891.25
b)Determine
the amount, principal plus interest,
Sherry will pay the credit union at the end of the
30 months to pay off her loan.
Sherry will pay the credit union $7091.25 at the end of 30 months
11.2-9
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Discount Notes
In another type of loan, the discount note, the
interest is paid at the time the borrower receives
the loan.
The interest charged in advance is called the
bank discount.
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Example 4: True Interest Rate of
a Discount Note
Siegrid Cook took out a $500 loan using a 10%
discount note for a period of 3 months. Determine
a)the
interest she must pay to the bank on the
date she receives the loan.
$12.50
b) the net amount of money she receives from the
bank.
$487.50
c) the actual rate of interest for the loan.
r ≈ 0.1026
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actual rate of interest is about 10.3% rather than the quoted 10%
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11.3 What You Will Learn
Compound Interest
Present Value
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Investments
An investment is the use of money or capital for
income or profit.
In a fixed investment, the amount invested as
principal is guaranteed and the interest is
computed at a fixed rate.
In a variable investment, neither the principal
nor the interest is guaranteed.
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Compound Interest
Interest that is computed on the principal
and any accumulated interest is called
compound interest.
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Compound Interest Formula

r
A = p 1 + 
n

nt
A is the amount that accumulates in the account
p is the principal
r
is the annual interest rate as a decimal
t
is the time in years
n is the number of compound periods per year
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Example 2: Using the Compound
Interest Formula
Kathy Mowers invested $3000 in a savings
account with an interest rate of 1.8%
compounded monthly. If Kathy makes no other
deposits into this account, determine the
amount in the savings account after 2 years.
The amount in the account after 2 years would be about $3109.88.
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Present Value Formula
p=
A

r
 1 + n 
nt
p is the present value, or principal to invest now
A is the amount to be accumulated in the account
r is the annual interest rate as a decimal
n is the number of compound periods per year
t is the time in years
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Example 5: Savings for College
Will Hunting would like his daughter to attend
college in 6 years when she finishes high school.
Will would like to invest enough money in a
certificate of deposit (CD) now to pay for his
daughter’s college expenses. If Will estimates
that he will need $30,000 in 6 years, how much
should he invest now in a CD that has a rate of
2.5% compounded quarterly?
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$25,833.30
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11.4 What You Will Learn
Fixed Installment Loans
Open-End Installment Loan
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Installments
A fixed installment loan is one on which you
pay a fixed amount of money for a set number
of payments.
Examples: college tuition loans, loans for cars,
boats, appliances, furniture, etc.
They are usually repaid in 24, 36, 48 or 60
months.
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Installments
An open-ended installment loan is a loan on
which you can make variable payments each
month.
Example: credit cards
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Truth in Lending Act in 1968
This law requires that the lending institution tell
the borrower two things:
The annual percentage rate (APR) is the true
rate of interest charged for the loan.
The total finance charge is the total amount of
money the borrower must pay for borrowing the
money: interest plus any additional fees charged.
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Total Installment Price
The total installment price is the sum of
all the monthly payments and the down
payment, if any.
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Table 11.2
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Installment Payment Formula
m=
r
p 
 n

r
1 − 1 + 
n

− n⋅t
m is the installment payment
p
is the amount financed
r
is the APR as a decimal
n
is the number of payments per year
t
is the time in years
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Example 2: Using the Installment
Payment Formula
Kristin Aiken wishes to purchase new window
blinds for her house at a cost of $1500. The home
improvement store has an advertised finance
option of no down payment and 6% APR for 24
months.
Determine Kristin’s monthly payment.
Kristin’s monthly payment would be $66.48.
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Repaying an Installment Loan
Early
By paying off a loan early, one is not
obligated to pay the entire finance charge.
The amount of the reduction of the finance
charge from paying off a loan early is called
the unearned interest.
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Repaying an Installment Loan
Early
Two methods are used to determine the finance
charge when you repay an installment loan early.
•
•
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The actuarial method uses the APR tables.
The rule of 78s does not use the APR tables, is
less frequently used, and is outlawed in much
of the country.
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Actuarial Method for Unearned
Interest
n ⋅ P ⋅V
u=
100 + V
u
is unearned interest
n
is # of remaining monthly payments
P
is the monthly payment
V
is the value from the APR table for the # of
remaining payments
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Example 5: Using the Actuarial
Method
Tino Garcia borrowed $9800 to purchase a classic
1966 Ford Mustang. The APR is 7.5% and there
are 48 payments of $237. Instead of making his
30th payment of his 48-payment loan, Tino
wishes to pay his remaining balance and
terminate the loan.
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Example 5: Using the Actuarial
Method
a) Use the actuarial method to determine how
much interest Tino will save (the unearned
interest, u) by repaying the loan early.
n = 18, P = $237, V = $6.04, u ≈ $242.99
Tino will save $242.99 in interest by repaying the loan early,
by actuarial method
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Example 5: Using the Actuarial
Method
b) What is the total amount due to pay
off the loan early on the day he
makes his final payment?
Solution
Remaining balance (exclude unearned interest):
18(237) – 242.99 = $4,023.01
Total amounts in final payment (remaining
balance plus 30th payment):
4,023.01 + 237 = $4,260.01
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Open-End Installment Loans
A credit card is a popular way of
making purchases or borrowing
money. Typically, credit card accounts
report:
*These rates vary with different credit card accounts and localities.
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Open-End Installment Loans
Typically, credit card monthly statements contain the
following information:
balance at the beginning of the period
balance at the end of the period (or new balance)
the transactions for the period
statement closing date (or billing date)
payment due date
the minimum payment due
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Open-End Installment Loans
For purchases, there is no finance or interest charge if
there is no previous balance due and you pay the entire
new balance by the payment due date.
The period between when a purchase is made and when
the credit card company begins charging interest is called
the grace period and is usually 20 to 25 days.
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Open-End Installment Loans
However, if you use a credit card to borrow
money, called a cash advance, there
generally is no grace period and a finance
charge is applied from the date you
borrowed the money until the date you
repay the money.
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Average Daily Balance
Many lending institutions use the average daily
balance method of calculating the finance charge
because they believe that it is fairer to the
customer.
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Example 8: Using the Average
Daily Balance Method
The balance on Min Zeng’s credit card
account on July 1, the billing date, was
$375.80. The following transactions occurred
during the month of July.
July
July
July
July
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5
10
18
28
Payment
Charge: Toy store
Charge: Garage
Charge: Restaurant
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$150.00
$74.35
$123.50
$42.50
Example 8: Using the Average
Daily Balance Method
a) Determine the average daily balance for
the billing period.
(i) find the balance by date
July
July
July
July
July
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1
5
10
18
28
$375.80
$375.80
$225.80
$300.15
$423.65
– $150.00
+ $74.50
+ $123.50
+ $42.50
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=
=
=
=
$225.80
$300.15
$423.65
$466.15
Example 8: Using the Average
Daily Balance Method
Solution
(ii)
Find the number of days that the balance did
not change between each transaction. Count
the first day in the period but not the last day.
(iii)
Multiply the balance due by the number of days
the balance did not change.
(iv)
Find the sum of the products.
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Example 8: Using the Average
Daily Balance Method
Solution
7/1
7/5
7/10
7/18
7/28
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$375.80
$225.80
$300.15
$423.65
$466.15
4
5
8
10
4
(375.80)(4) = $1503.20
(225.80)(5) = $1129.00
(300.15)(8) = $2401.20
(423.65)(10)=$4236.50
(466.15)(4) = $1864.60
31
Sum = $11,134.50
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Example 8: Using the Average
Daily Balance Method
Solution
(v) Divide the sum by the number of
days
$11,134.50 ÷ 31 = $359.18
The average daily balance is $359.18.
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Example 8: Using the Average
Daily Balance Method
b) Determine the finance charge to be
paid on August 1, Min’s next billing
date. Assume that the interest rate
is 1.3% per month.
Solution
Use the simple interest formula:
i = prt
= $359.18 × 0.013 × 1
≈ $4.67
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Example 8: Using the Average
Daily Balance Method
c) Determine the balance due on
August 1.
The balance due on August 1
= the balance (at last purchase) + finance charge
= $466.15 + $4.67
= $470.82.
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11.6 What You Will Learn
Ordinary Annuities
Sinking Funds
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Annuity
An annuity is an account into which,
or out of which, a sequence of
scheduled payments is made.
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Ordinary Annuity
An annuity into which equal payments are made
at regular intervals, with the interest compounded
at the end of each interval and with a fixed
interest rate for each compounding period, is
called an ordinary annuity or fixed annuity.
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Ordinary Annuity
The amount of money that is present in an
ordinary annuity after t years is known as the
accumulated amount or the future value of an
annuity.
Two methods:
•Spread sheet
•Formula
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Ordinary
Annuity
1st half of
table
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Ordinary
Annuity
2nd half
of table
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Ordinary Annuity
Another way: the ordinary annuity formula
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Ordinary Annuity Formula
nt


r
p   1+  − 1
n

 
A=
r
n
A: accumulated amount (A is the unknown)
p: dollars in each payment
n: times of payments per year
t: years
r: annual interest rate
compounded at the end of each payment period
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Example 1: Using the Ordinary
Annuity Formula
Bill and Megan Lutes are depositing $250 each
quarter in an ordinary annuity that pays 4%
interest compounded quarterly. Determine the
accumulated amount in this annuity after 35
years.
There will be about $75,677.48 in Bill and Megan’s
annuity after 35 years.
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Timely Tip
An ordinary annuity is used when you wish to
determine the accumulated amount.
A sinking fund is used when you wish to determine
how much money an investor must invest each
period to reach an accumulated amount at a
specific time.
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Sinking Fund
A sinking fund is a type of annuity in
which the goal is to save a specific
amount of money in a specific amount
of time.
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Sinking Fund Payment Formula
p=
r
A 
 n

r
 1+ n 
nt
−1
p: dollars in each payment (p is the unknown)
A: accumulated amount
n: times of payments per year
t: years
r: annual interest rate
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Example 2: Using Sinking Fund
Payment Formula
Abigail and John Clayton would like to have $55,000 in 10
years to remodel their home. The Claytons decide to invest
monthly in a sinking fund that pays 3.3% interest
compounded monthly. How much should the Claytons
invest in the sinking fund each month to accumulate
$55,000 in 10 years?
Clayton’s need to invest $387.49 each month to
accumulate $55,000 in 10 years.
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