Section 8.3 Compound Interest Math in Our World

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.  Compute the periodic payment needed to meet an investment goal.

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

EXAMPLE 1 Comparing Simple and Compound Interest Suppose that $5,000 is invested for 3 years at 8%. (a) Find the amount of simple interest. (b) Find the compound interest if interest is calculated once per year. SOLUTION (a) Using the formula I = Prt with P = $5,000, r = 0.08, and t = 3, we get I = $5,000 x 0.08 x 3 = $1,200 The amount of simple interest earned over 3 years is $1200.

EXAMPLE 1 Comparing Simple and Compound Interest SOLUTION (b) First year: We have P = $5,000, r = 0.08 and t = 1: I = Prt = $5,000 x 0.08 x 1 = $400 The interest for the first year is $400. Second year: At the beginning of the second year, the account now contains $5,400, so we use this as principal for the second year. The rate and time remain the same. I = Prt = $5,400 x 0.08 x 1 = $432 The interest for the second year is $432.

EXAMPLE 1 Comparing Simple and Compound Interest SOLUTION Third year: The principal is now $5,400 + $432 = $5,832. I = Prt = $5,832 x 0.08 x 1 = $ The interest for the third year is $ The total interest for three years is $400 + $432 + $ = $1, This is almost a hundred dollars more than with simple 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).

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, compounded quarterly at 3% for 5 years. SOLUTION Quarterly means 4 times a year, so n = 4. P = $7,000.00, r = 3% = 0.03, t = 5

EXAMPLE 2 Computing Compound Interest SOLUTION To find the interest, subtract the principal from the future value. I = $8, – $7, = $1, The interest is $1,

EXAMPLE 3 Computing Compound Interest Find the interest on $11,000 compounded daily at 5% for 6 years. Assume a 365-day year. SOLUTION P = $11,000, r = 5% = 0.05, n = 365, t = 6 To find the interest, subtract the principal from the future value. I = $14, – $11, = $3,848.14

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 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.

EXAMPLE 4 Finding Effective Interest Rate Find the effective interest rate when the stated rate is 4% and the interest is compounded semiannually. SOLUTION Let r = 0.04 (rate is 4%) and n = 2 (compounded semiannually) and then substitute into the formula. The effective rate is 4.04%.

EXAMPLE 5 Comparing the Effective Rate of Two Investments Which savings account is a better investment: 6% compounded quarterly or 6.2% compounded semiannually? SOLUTION Find the effective rates of both accounts and compare them.

EXAMPLE 5 Comparing the Effective Rate of Two Investments SOLUTION The 6.2% semiannual investment gives an effective rate of 6.3%, while 6% quarterly gives an effective rate of 6.14%, so 6.2% semiannually is a better investment.

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.

EXAMPLE 6 Finding Future Value of an Annuity Find the future value of an annuity where a $500 payment is made annually for 3 years at 6%. SOLUTION The interest rate is 6% and the payment is $500 each year for 3 years. I. End of the first year $500 (payment)

EXAMPLE 6 Finding Future Value of an Annuity SOLUTION II. End of the second year The $500 collected 6% interest and a $500 payment is made; the value of the annuity at the end of the second year is $500(0.06) = $ 30 Interest. $500 Principal paid at end of first year. + $500 Payment at the end of the second year. $1,030

EXAMPLE 6 Finding Future Value of an Annuity SOLUTION III. End of the third year During the third year, the $1,030 earns 6% interest and a payment of $500 is made at the end of the third year. The annuity is worth $ 1,030(0.06) = $61.80 Interest. $1, Principal at end of second year. + $ Payment at the end of third year. $1, The future value of the annuity at the end of the three years is $1,

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.

EXAMPLE 7 Finding the Future Value of an Annuity SOLUTION R = $800, r = 5% = 0.05, n = 2 (semiannual) and t = 4 This is future value of the annuity at the end of 4 years.

Annuity Payments Formula for Finding Regular Annuity Payments Needed to Reach a Goal 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 8 Finding the Monthly Payment for an Annuity Suppose you’ve always dreamed of opening your own tattoo parlor, and decide it’s time to do something about it. A financial planner estimates that you would need a $35,000 initial investment to start the business, and you plan to save that amount over the course of 5 years by investing in an annuity that pays 7.5% compounded weekly. How much would you need to invest each week?

EXAMPLE 8 Finding the Monthly Payment for an Annuity We know the following values: A = 35,000, r = 7.5% = 0.075, t = 5, and n = 52. SOLUTION A payment of $ per week would be necessary to save the required $35,000.