March 25 2010 Chapters 21 amp 22 Chapter 21 Arithmetic Growth amp Simple Interest Geometric Growth amp Compound Interest A Model for Saving Present Value Chapter 22 Simple Interest Compound Interest ID: 613776
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Savings & Borrow ModelsMarch 25, 2010
Chapters 21 & 22Slide2
Chapter 21Arithmetic Growth & Simple InterestGeometric Growth & Compound Interest
A Model for Saving
Present Value Chapter 22Simple InterestCompound InterestConventional LoansAnnuitiesSlide3
Definitions:Principal—initial balance of an account
Interest—amount added to an account at the end of a specified time period
Simple Interest—interest is paid only on the principal, or original balanceArithmetic Growth & Simple InterestSlide4
Interest (I)
earned in terms of
t years, with principal P and annual rate r:
I=
Prt
Arithmetic growth (also referred to as linear growth) is growth by a constant amount in each period.
Simple InterestSlide5
Simple Interest on a Student LoanP
= $10,000
r = 5.7% = 0.057t = 1/12 yearI
for one month = $47.50
Exercise #1Slide6
Compound interest—interest that is paid on both principal and accumulated interest
Compounding period—time elapsing before interest is paid; i.e. semi-annually, quarterly, monthly
Geometric Growth & Compound InterestSlide7
Effective Rate & APYEffective rate is the rate of simple interest that would realize exactly as much interest over the same length of time
Effective rate for a year is also called the annual percentage yield or APY
Rate Per Compounding PeriodFor a given annual rate r
compounded
m times
per year, the rate per compound period is
Periodic rate =
i
= r/m
Geometric Growth & Compound InterestSlide8
For an initial principal P
with a periodic interest rate
i per compounding period grows after n compounding periods to:A=P(1+i)
n
For an annual rate, an initial principal
P
that pays interest at a nominal annual rate
r
, compounded
m
times per year, grows after
t
years to:
A=P(1+r/m)
mt
Compound InterestSlide9
A amount accumulated
P
initial principalr nominal annual rate of interest
t
number of years
m
number of compounding periods per year
n =
mt
total number of compounding periods
i
= r/m
interest rate per compounding period
Geometric growth (or exponential growth) is growth proportional to the amount present
Notation For SavingsSlide10
Effective Rate and APY
Effective rate = (1+
i)n-1
APY = (1 +
r/m
)
m
-1
Exercise #2
APY = 6.17%Slide11
FormulasGeometric Series
1 +
x +x2 +x3 + …
+x
n
-1 = (
x
n
-1)/(x
-1)
Annuity—a specified number of (usually equal) periodic payments
Sinking Fund—a savings plan to accumulate a fixed sum by a particular date, usually through equal periodic deposits
A Model for SavingSlide12
Present value—how much should be put aside now, in one lump sum, to have a specific amount available in a fixed amount of time
P = A/(1+i)
n= A/(1+r/m)mt
Exercise #3
What amount should be put into the CD?
Present ValueSlide13
When borrowing with simple interest, the borrower pays a fixed amount of interest for each period of the loan, which is usually quoted as an annual rate.
I=
PrtTotal amount due on loanA=P(1+rt)
Simple InterestSlide14
Compound Interest Formula Principal
P
is loaned at interest rate I per compounding period, then after n compounding periods (with no repayment) the amount owed is
A=P(1+i)
n
When loaned at a nominal annual rate
r
with
m
compounding periods per year, after
t
years
A=P(1+r/m)
mt
A nominal rate is any state rate of interest for a specified length of time and does not indicate whether or how often interest is compounded.
Compound InterestSlide15
First month’s interest is 1.5% of $1000, or 0.015 ∙ $1000 = $15Second month’s interest is now 0.015 ∙ $1015 = $15.23
After 12 months of letting the balance ride, it has become
(1.015)12 ∙ $1000 = $1195.62
Annual Percentage Rate (APR) is the number of compounding periods per year times the rate of interest per compounding period:
APR =
m ∙
i
Exercise #4Slide16
Loans for a house, car, or college expenses
Your payments are said to
amortize (pay back) the loan, so each payments pays the current interest and also repays part of the principalExercise #5
P
= $12,000
i
=
0.049/12 n
= 48
monthly payment = $275.81
Conventional LoansSlide17
An annuity is a specified number of (usually equal) periodic payments.
Exercise #6
d = $1000 r
= 0.04
m
= 12
t
= 25P = $189,452.48
AnnuitiesSlide18
8th
Edition
Chapter 21225Chapter 225
Discussion & Homework