Introducing Simple and Compound Interest

This topic explains simple interest and compound interest through a series of problems and examples. Compounding continuously and the annual percentage rate is also worked on. Attention is given to the problem of finding the doubling time for an investment.

Definition (Simple Interest and Future Value) If a sum of money (called the principal) is invested for a period of time at an interest rate per period, the simple interest is given by the formula: and the future value of the investment is

Example (Future Value for Simple Interest) If $21,200 is invested at an annual simple interest rate of 5%, what is the future value of the investment after 2 years?
The future value is given by the formula and since and we have

Example (Interest for Simple Interest) If $7,700 is invested for 5 years at an annual simple interest rate of 15%, how much interest is earned?
The interest earned is where and so we have

Example (Principal for Simple Interest) A firm buys 15 file cabinets at $166.23 each, with the bill due in 90 days. How much must the firm deposit now to have enough to pay the bill if money is worth 6% per year? Use 360 days in a year.
The future value is We are looking for the principal, and We use the formula and we have and solving for we get

Example (Doubling Time for Simple Interest) If $5000 is invested at 8% annual simple interest, how long does it take to double to $10,000?
The future value is given by the formula and we are given a value of We are asked to find when and We have

years.

Definition (Periodic Compounding Interest) If dollars is invested for years at a nominal interest rate componded times per year, then the total number of compounded periods is and the interest rate per period is and the future value is or

Example (Future Value for Compounding Periocially) Find the future value if $3500 is invested for 6 years at 8% compounded quarterly.
The future value is given by the formula where and so we have

Example (Interest for Compounding Periocially) Find the interest that will be earned if $5000 is invested for 3 years at 10% compounded semiannually.
The interest earned is the future value minus the principal. So we find the future value first. The future value is given by where and so we have

Therefore, the interest earned is

Example (Principal for Compounding Periocially) What present value amounts to $100,000 if it is invested for 10 years at 8% compounded quarterly?
The present value can be found using the formula where the future value and so we have

Example (Doubling Time for Componding Periocially) How long in years would $700 have to be invested at 11.9% compounded monthly to have $1,400?
The future value is and can be found using the formula where and so we have

years.

Definition (Continuous Compounding Interest) If dollars is invested for years at an interest rate compounded continuously, then the future value is given by

Example (Future Value for Compounding Continuously) What lump sum do parents need to deposit in an account earning 9%, compounded continuously, so that it will grow to $40,000 for their daughter's college tuition in 18 years?
The future value is $40,000 and is given by the formula where and and so we have

Example (Interest for Compounding Continuously) Which investment will earn more money, a $1000 investment for 6 years at 8% componded annually, or a $1000 investment for 6 years compounded continuously?
The investment that is compounding annually will have future value of where and which is The investment that is compounding continuously will have future value where and which is Thus, the investment which is compounding continuously is the better investment.

Example (Principal for Compounding Continuously) What present value needs to be deposited to have $20,000 in 3 years with an investment that is compounded continuously at 4%?
The future value is 20000 and is given by the formula where and and so we have

Example (Doubling Time for Compounding Continuously) (a) How long in years would $700 have to be invested at 12.3%, componded continuously, to have
    The future value is and is given by the formula where and and so we have

years

    (b) Find the doubling time for an investment with interest rate and principal where is in years.
    The doubling time is given by the future value formula where is the present value, is the interest rate, and is the time in years, so we have

Definition (Annual Percentage Yield) If is the number of compounding periods per year, then is the interest rate per period and if is the annual interest rate for an investment, then the annual percentage yield is defined by the formula

For compounded continuously invesment the A.P.Y. is defined by the formula

Example (Annual Percentage Yield) Suppose there are three investements to invest in (a) one at 10% compounded annually, (b) another at 9.8% compounded quarterly, and (c) a third investment at 9.65% compounded continuously. Which investment is best?
For the first investment and and so will have A.P.Y. For the second investment we have and and so we have A.P.Y. For the last investment we have A.P.Y. and so the best investment is the second.

Example (Interest Problems) (a) What is the present value of an investment at 6% annual simple interest if it is worth $832 in 8 months?
    The future value is 832 and is given by where and and so we have


    (b) How much more interest will be earned if $5000 is invested for 6 years at 7% compounded continuously, instead of at 7% compounded quarterly?
    If we use compounding continuously then the future value is where and and so we have Thus the interest earned is If we use compounding quarterly then the future value is given by where and and so we have future value of Thus for compounding quarterly we have interest earned as Therefore, the first investment is better by