Savings & Borrow Models

March 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

A amount 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