chapters 21 & 22
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Chapters 21 & 22. 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. - PowerPoint PPT PresentationTRANSCRIPT
Savings & Borrow ModelsMarch 25, 2010
Chapters 21 & 22
Chapter 21Arithmetic Growth & Simple
InterestGeometric Growth & Compound
InterestA Model for SavingPresent Value Chapter 22Simple InterestCompound InterestConventional LoansAnnuities
Definitions:Principal—initial balance of an accountInterest—amount added to an account at the
end of a specified time periodSimple Interest—interest is paid only on the
principal, or original balance
Arithmetic Growth & Simple Interest
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
Simple Interest on a Student LoanP = $10,000r = 5.7% = 0.057t = 1/12 yearI for one month = $47.50
Exercise #1
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 Interest
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 isPeriodic rate = i = r/m
Geometric Growth & Compound Interest
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 Interest
Aamount accumulatedP initial principalr nominal annual rate of interestt number of yearsm number of compounding periods per yearn = mt total number of compounding periodsi = r/m interest rate per compounding period
Geometric growth (or exponential growth) is growth proportional to the amount present
Notation For Savings
Effective Rate and APYEffective rate = (1+i)n-
1
APY = (1 +r/m)m-1
Exercise #2APY = 6.17%
FormulasGeometric 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
A Model for Saving
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 #3What amount should be put into the CD?
Present Value
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 loanA=P(1+rt)
Simple Interest
Compound Interest FormulaPrincipal 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 Interest
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
Exercise #4
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 #5P = $12,000i = 0.049/12n = 48monthly payment = $275.81
Conventional Loans
An annuity is a specified number of (usually equal) periodic payments.
Exercise #6d = $1000r = 0.04m = 12t = 25
P = $189,452.48
Annuities
8th EditionChapter 21225Chapter 225
Discussion & Homework