Math in Our World Section 8.3 D1 Compound Interest

Learning Objectives Compute compound interest. Compute the effective interest rate of an investment. Compare the effective rate of two investments. Find the future value of an annuity.

Compound Interest When interest is computed on the principal and any previously earned interest, it is called compound interest.

Compound Interest When interest is calculated once each year, we say that it is compounded yearly. In many cases, interest is computed at more frequent intervals than that. It can be compounded semiannually (twice a year), quarterly (4 times a year), monthly (12 times a year), or even daily (every day).

EXAMPLE 1 Finding Compound Interest the Long Way Suppose that $7,000 is invested for 5 years at 3%. 1st year - I=$7,000x.03x1 = $210 A= $7,210 2nd year – I=$7,210x.03x1 = $216.30 A= $7,426.30 3rd year – I=$7,426.30x.03x1 = $222.79 A= $7,649.09 4th year – I=$7,649.09x.03x1 = $229.47 A= $7,878.56 5th year – I=$7,878.56x.03x1 = $236.36 A= $8,114.92

Compound Interest Formula for Computing Compound Interest where A is the future value (principal + interest) r is the yearly interest rate in decimal form n is the number of times per year the interest is compounded t is term of the investment in years

EXAMPLE 2 Computing Compound Interest Find the interest on $7,000.00 compounded quarterly at 3% for 5 years.

Effective Interest Rate The effective rate (also known as the annual yield or nominal rate) is the simple interest rate which would yield the same future value over 1 year as the compound interest rate. Formula for Effective Interest Rate where E = effective rate r = interest rate per year (i.e., stated rate) n = number of periods per year the interest is calculated

EXAMPLE 4 Finding Effective Interest Rate Find the effective interest rate when the stated rate is 4% and the interest is compounded semiannually.

Annuities An annuity is a savings investment for which an individual or business makes the same payment each period (i.e., annually, semiannually, or quarterly) into a compound- interest account where the interest does not change during the term of the investment. The total amount accumulated (payments plus interest) is called the future value of the annuity. Annuities are set up by individuals to pay for college expenses, vacations, or retirement.

Annuities Formula for Finding the Future Value of an Annuity where A is the future value of the annuity R is the regular periodic payment r is the annual interest rate n is the number of payments made per year t is the term of the annuity in years

EXAMPLE 7 Finding the Future Value of an Annuity Find the future value of an annuity when the payment is $800 semiannually, the interest rate is 5% compounded semiannually, and the term is 4 years.