Unit 4B

The Power of Compounding

PERCENTAGES

r % means r percent rrr 01.01.0100

1

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)

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.

ANNUAL PERCENTAGE RATE

An annual percentage interest rate (APR) of r% on a principal P means 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)

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

COMPOUND INTEREST FORMULA FOR INTEREST PAID ONCE A YEAR

A = P(1 + APR)Y

where

A = accumulated balance after Y yearsP = starting principle

APR = annual percentage rate (as a decimal)Y = number of years

COMPOUND INTEREST FORMULA FOR INTEREST PAID n TIMES A YEAR

)(APR

1Yn

nPA

where

A = accumulated balance after Y yearsP = starting principle

APR = annual percentage rate (as a decimal)n = number of compounding periods per yearY = number of years

COMPOUND INTEREST FOR CONTINUOUS COMPOUNDING

A = P e (APR × Y)

where

A = accumulated balance after Y yearsP = 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 . . . .

ANNUAL PERCENTAGE YIELD

• 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.

The annual percentage yield (APY) is the actual percentage by which a balance increases in one year.

COMPUTING THE ANNUAL PERCENTAGE YIELD (APY)

100principalstarting

increaseabsoluteAPY