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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.

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Chapters 21 & 22
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  1. Chapters 21 & 22 Savings & Borrow Models March 25, 2010

  2. 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

  3. Arithmetic Growth & Simple Interest 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

  4. 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.

  5. Exercise #1 Simple Interest on a Student Loan • P = $10,000 • r = 5.7% = 0.057 • t = 1/12 year • I for one month = $47.50

  6. Geometric Growth & Compound Interest 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

  7. Geometric Growth & Compound Interest 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

  8. 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

  9. Notation For Savings A amount accumulated P initial principal r 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

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

  11. A Model for Saving 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

  12. Present Value 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?

  13. Simple Interest 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)

  14. Compound Interest 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.

  15. Exercise #4 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

  16. Conventional Loans • 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

  17. Annuities 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

  18. Discussion & Homework 8th Edition Chapter 21 • 2 • 25 Chapter 22 • 5