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% = 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

Aamount accumulated Pinitial principal rnominal annual rate of interest t number of years mnumber 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

Effective rate = (1+i) n -1 APY = (1 +r/m) m -1 Exercise #2 APY = 6.17%

Formulas Geometric Series 1 + x +x 2 +x 3 + … +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

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 ∙ $1000 = $15 Second month’s interest is now ∙ $1015 = $15.23 After 12 months of letting the balance ride, it has become (1.015) 12 ∙ $1000 = $ 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

8 th Edition Chapter Chapter 22 5