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Simple Interest

Definition (Interest Earned) The difference between the accumulated value of an investment after a period of time and the initial amount invested (principal) is called the interest earned during that period.

Example (Interest Earned)An initial deposit of 1,000 is made into an investment account. No withdrawals are made and no other principal is invested. At the end of three years, the account is worth 1,300. The interest earned during the three-year period is

Definition (Effective Rate of Interest) The effective rate of interest is the amount of money that one unit of principal will earn during one investment period. Alternatively, it is the ratio of the amount of interest earned during the period to the amount of principal invested at the beginning of the period.

Example (Effective Rate of Interest) (a) In Example 1.3, the effective rate of interest earned over the three-year period can be calculated as
    (b) An initial deposit of 500 accumulates to 520 at the end of one year and 550 at the end of the second year. The effective rate of interest for the first year is   The effective rate of interest for the second year is The effective rate of interest for the two-year period is
    Note that it was assumed that the interest earned in the first year was re-invested during the second year, so that the principal amount at the beginning of the second year was 520. In part (a) no such assumption is needed. Also, note that the effective rate of interest can be calculated for any time period; in part (b) it was calculated separately for the first year, second year, and the two-year period. Note that the and are annual rates while the is a two-year rate. Topic 4, Equations of Value, will discuss methods of converting the two-year rate to an annual rate.

Definition (Simple Interest) A pattern of interest accrual such that the amount of interest earned in each investment period is constant is called simple interest. If is the initial investment and is the constant interest amount, the accumulated value at the end of investment periods is given by

Example (Simple Interest) (a) If 1050 is invested for three years at 4%, find the accumulated value and total interest earned at the end of the period using simple interest.
    Under simple interest, the 4% simple interest rate determines the constant interest amount by The accumulated value after 3 years is The total interst earned in the three year period is Note that
    (b) If 100 is invested at 3% simple interest, find the accumulated value after 3 years and the effective rates of interest in the third and fourth year.
    The accumulated value after three years is given by the formula where and Therefore,
    


     The effective rate of interest in the third year is the amount of interest in the third year divided by the accumulated amount at the end of the second year.
     


     The effective rate of interest in the fourth year is the amount of interest in the fourth year divided by the accumulated amount at the end of the third year.
     


    (c) If 1115 is invested to earn a constant 44.6 each year, how many years will it take for the account value to double?
    The situation described is simple interest, since the interest amount is constant. We want to solve for given that the accumulated value is 2230.
    




    (d) If 2130 is invested to earn a constant rate of 4%, what is the accumulated value after 4 years? How many whole years will it take for the account value to become greater than 3000?
    The accumulated value after 4 years is given by the formula where and Substituting, we get
    


    To find the number of years it will take for the account value to grow to over 3000, we must solve the equation
    




therefore, the number of whole years for the account value to grow to over 3000 is 11.

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