Learning Objectives for Sections 3.1-3.2 Simple & Compound Interest

Learning Objectives for Sections 3.1-3.2 Simple & Compound Interest After this lecture, you should be able to • Compute simple interest using the simple interest formula. • Solve problems involving investments and the simple interest formula. • Compute compound interest.

Some Preliminary Notes • Financial institutions often use 360 days for one year when computing time. • Time must be in terms of years to use in the formulas. • All rates (%) must be converted to decimals to use in formulas. • When an answer is rounded, use the symbol instead of =. • We will round to the nearest cent for dollar amounts, unless otherwise stated. • Try to avoid rounding until the final answer.

Conversions: Time Periods • Example 1: Convert the given time periods into years: • 180 days b) 120 days c) 3 quarters • d) 7 months e) 60 days f) 26 weeks

Conversions: Percents to Decimals • Example 2: Convert the given percents to decimals: • 4.5% b) .32% c) 112%

Conversions: Decimals to Percents • Example 3: Convert the given decimals to percents: • 0.06 b) 5 c) 0.11

Simple Interest Formula Simple Interest Formula where I = interest P = principal (amount invested or amount of loan) r = annual simple interest rate (as a decimal) t = time in years

An Example Example 4: Find the interest on a boat loan of $5,000 at 16% for 8 months. Example 5: What is the total amount to be paid back on the boat loan in Example 4?

Total Amount to Be Paid Back • In general, the future value (amount) is given by the following equation: A = Principal + Interest A = P + I A = P + Prt A = P (1 + rt)

Another Example Example 6: Find the total amount due on a loan of $600 at 16% interest at the end of 15 months.

Another Example Example 7:A loan of $10,000 was repaid at the end of 6 months. What amount (principal and interest) was repaid, if a 6.5% annual rate of interest was charged?

Application Example 8: A department store charges 18.6% interest (annual) for overdue accounts. How much interest will be owed on a $1,080 account that is 3 months overdue?

Example 10:What is the purchase price of a 26-week T-bill with a maturity value of $1,000 that earns an annual interest rate of 4.903%? Purchase Price of a Note

Compound Interest • Compound interest: Interest paid on interest reinvested. • Compound interest is always greater than or equal to simple interest in the same time period, given the same annual rate.

Compounding Periods The number of compounding periods per year (m): • If the interest is compounded annually, then m = _______ • If the interest is compounded semiannually, then m = _______ • If the interest is compounded quarterly, then m = _______ • If the interest is compounded monthly, then m = _______ • If the interest is compounded daily, then m = _______

Example 1: Suppose a principal of $1 was invested in an account paying 6% annual interest compounded monthly. How much would be in the account after one year? Example The annual interest rate is 6%, so the monthly interest rate would be: In general, we can find the rate per compounding period as: Continued on next slide.

Solution • Solution: Using the Future Value with simple interest formula A = P (1 + rt) we obtain the following amount: • after one month: • after two months: • after three months: After 12 months, the amount is: ________________________. With simple interest, the amount after one year would be _______. The difference becomes more noticeable after several years.

Graphical Illustration ofCompound Interest The growth of $1 at 6% interest compoundedmonthly compared to 6% simple interest over a 15-year period. The blue curve refers to the $1 invested at 6% simple interest. Dollars The red curve refers to the $1 at 6% being compounded monthly. Time (in years)

General Formula: Compound Interest The formula for calculating the Future Amount with Compound Interest is Where A is the future amount, P is the principal, r is the annual interest rate as a decimal, m is the number of compounding periods in one year, and tis the total number of years.

Example 2a: Find the amount to which $1,500 will grow if compounded quarterly at 6.75% interest for 10 years. Example 2b:Compare your answer from part a) to the amount you would have if the interest was figured using the simple interest formula. Example

Changing the number of compounding periods per year Example 3: To what amount will $1,500 grow if compounded daily at 6.75%interest for 10 years?

Effect of Increasing the Number of Compounding Periods • If the number of compounding periods per year is increased while the principal, annual rate of interest and total number of years remain the same, the future amount of money will increase slightly.

Example Example 4: If $20,000 is invested at 4% compounded monthly, what is the amount after a) 5 years b) 8 years?

Which is Better? Example 5: Which is the better investment and why: 8% compounded quarterly or 8.3% compounded annually?

Inflation Example 6: If the inflation rate averages 4% per year compounded annually for the next 5 years, what will a car costing $17,000 now cost 5 years from now?

Learning Objectives for Sections 3.1-3.2 Simple & Compound Interest