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Unit 4B

Unit 4B. The Power of Compounding. r % means r percent. PERCENTAGES. r % of P means 0.01 r × P = 0.1 rP. To increase P by r % means P + ( r % of P ) = P + 0.01 rP = P (1 + 0.01 r ). To decrease P by r % means P − ( r % of P ) = P − 0.01 rP

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Unit 4B

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  1. Unit 4B The Power of Compounding

  2. r % means r percent PERCENTAGES r % of P means 0.01r× P = 0.1rP To increase P by r % means P + (r % of P) = P + 0.01rP = P(1 + 0.01r) To decrease P by r % means P− (r % of P) = P − 0.01rP = P(1 − 0.01r)

  3. DEFINITIONS The principal in a financial formula is the balance upon which interest is paid. Simple interest is interest paid only on the original principal, and not on any interest added at later dates. Compound interest is interest paid both on the original principal and on all interest that has been added to the original principal.

  4. ANNUAL PERCENTAGE RATE An annual percentage interest rate (APR) of r% on a principal Pmeans multiplying the amount by (1 + .01r ). If we start with a principal P, then after 1 year, the accumulated balance becomes A1 = P(1 + 0.01r)

  5. After 2 years, the balance becomes A2 = P(1 + 0.01r)2 After 3 years, the balance becomes A3 = P(1 + 0.01r)3 Hence, after Y years, the balance becomes AY = P(1 + 0.01r)Y

  6. COMPOUND INTEREST FORMULA FOR INTEREST PAID ONCE A YEAR A = P(1 + APR)Y where A = accumulated balance after Y years P = starting principle APR = annual percentage rate (as a decimal) Y = number of years

  7. COMPOUND INTEREST FORMULA FOR INTEREST PAID n TIMES A YEAR where A = accumulated balance after Y years P = starting principle APR = annual percentage rate (as a decimal) n = number of compounding periods per year Y = number of years

  8. COMPOUND INTEREST FOR CONTINUOUS COMPOUNDING A = P e (APR ×Y) where A = accumulated balance after Y years P = starting principle APR = annual percentage rate (as a decimal) Y = number of years The number e is a special irrational number with a value of e≈ 2.71828128459045 . . . .

  9. ANNUAL PERCENTAGE YIELD The annual percentage yield (APY) is the actual percentage by which a balance increases in one year. • The APY is equal to the APR if interest is compounded annually; • it is greater than the APR if interest is compounded more than once a year; • and it is independent of the starting principal. • The annual percentage yield is also called the effective annual yield.

  10. COMPUTING THE ANNUAL PERCENTAGE YIELD (APY)

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