Transcript Chapters 21 & 22 - William & Mary Mathematics

Savings & Borrow Models
March 25, 2010
Chapter 21
 Arithmetic Growth & Simple Interest
 Geometric Growth & Compound
Interest
 A Model for Saving
 Present Value
Chapter 22
 Simple Interest
 Compound Interest
 Conventional Loans
 Annuities
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 balance
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 Interest on a Student Loan
 P = $10,000
 r = 5.7% = 0.057
 t = 1/12 year
 I for one month = $47.50
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
Effective Rate & APY
 Effective 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 Period
 For a given annual rate r compounded m times per
year, the rate per compound period is
Periodic rate = i = r/m
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
A
P
r
t
m
n = mt
i = r/m
amount accumulated
initial principal
nominal annual rate of interest
number of years
number of compounding periods per year
total number of compounding periods
interest rate per compounding period
Geometric growth (or exponential growth) is growth
proportional to the amount present
Effective rate = (1+i)n-1
APY = (1 +r/m)m-1
Exercise #2
APY = 6.17%
Formulas
 Geometric Series
 1 + x +x2 +x3 + … +xn-1 = (xn-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
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?
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=Prt
Total amount due on loan
A=P(1+rt)
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.
First month’s interest is 1.5% of $1000, or 0.015 ∙ $1000 = $15
Second 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
 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 principal
Exercise #5
P = $12,000
i = 0.049/12
n = 48
monthly payment = $275.81
An annuity is a specified number of (usually equal)
periodic payments.
Exercise #6
d = $1000
r = 0.04
m = 12
t = 25
P = $189,452.48
8th Edition
Chapter 21
 2
 25
Chapter 22
 5